Analysis is essentially about taking limits. The most basic type of a limit is a limit of a sequence of real numbers. We have already seen sequences used informally. Let us give the formal definition.
Definition2.1.1.
A sequence (of real numbers) is a function \(x \colon \N \to
\R\text{.}\) Instead of \(x(n)\text{,}\) we usually denote the \(n\)th element in the sequence by \(x_n\text{.}\) To denote a sequence we write 1
A sequence \(\{ x_n \}_{n=1}^\infty\) is bounded if the underlying function is bounded. That is, if there exists a \(B \in \R\) such that
\begin{equation*}
\abs{x_n} \leq B \qquad \text{for all } n \in \N.
\end{equation*}
In other words, the sequence \(\{x_n\}_{n=1}^\infty\) is bounded whenever the set \(\{ x_n : n \in \N \}\) is bounded. We similarly define the words bounded below and bounded above.
When we need to give a concrete sequence, we often give each term as a formula in terms of \(n\text{.}\) For example, \(\{ \nicefrac{1}{n} \}_{n=1}^\infty\) stands for the sequence \(1, \nicefrac{1}{2}, \nicefrac{1}{3}, \nicefrac{1}{4},
\nicefrac{1}{5}, \ldots\text{.}\) The sequence \(\{ \nicefrac{1}{n} \}_{n=1}^\infty\) is a bounded sequence (\(B=1\) suffices). On the other hand, the sequence \(\{ n \}_{n=1}^\infty\) stands for \(1,2,3,4,\ldots\text{,}\) and this sequence is not bounded (why?).
While the notation for a sequence is similar 2 to that of a set, the notions are distinct. For example, the sequence \(\bigl\{ {(-1)}^n \bigr\}_{n=1}^\infty\) is the sequence \(-1,1,-1,1,-1,1,\ldots\text{,}\) whereas the set of values, the range of the sequence, is just the set \(\{ -1, 1 \}\text{.}\) We write this set as \(\bigl\{ {(-1)}^n : n \in \N \bigr\}\text{.}\)
Another example of a sequence is the so-called constant sequence. That is a sequence \(\{ c \}_{n=1}^\infty = c,c,c,c,\ldots\) consisting of a single constant \(c \in \R\) repeating indefinitely.
Definition2.1.2.
A sequence \(\{ x_n \}_{n=1}^\infty\) is said to converge to a number \(x \in \R\) if for every \(\epsilon > 0\text{,}\) there exists an \(M \in \N\) such that \(\abs{x_n - x} < \epsilon\) for all \(n \geq M\text{.}\) The number \(x\) is called a limit of the sequence. If the limit \(x\) is unique, we write 3
\begin{equation*}
\lim_{n\to \infty} x_n \coloneqq x .
\end{equation*}
A sequence that converges is said to be convergent. Otherwise, we say the sequence diverges or that it is divergent.
Shortly, in Proposition 2.1.6 we will show that the limit \(x\) is always unique if it exists. It makes sense to talk about the limit of a sequence and we only need to show that the sequence converges to one number. For the next couple of examples, let us pretend we have already proved that limits are unique.
Intuitively, the limit being \(x\) means that eventually every number in the sequence is close to the number \(x\text{.}\) More precisely, we get arbitrarily close to the limit, provided we go far enough in the sequence. It does not mean we ever reach the limit. It is possible, and quite common, that there is no \(x_n\) in the sequence that equals the limit \(x\text{.}\) We illustrate the concept in Figure 2.1. In the figure we first think of the sequence as a graph, as it is a function of \(\N\text{.}\) Secondly, we also plot it as a sequence of labeled points on the real line.
Figure2.1.Illustration of convergence. On top, we show the first ten points of the sequence as a graph with \(M\) and the interval around the limit \(x\) marked. On bottom, the points of the same sequence are marked on the number line.
When we write \(\lim_{n\to\infty} x_n = x\) for some real number \(x\text{,}\) we are saying two things: First, that \(\{ x_n \}_{n=1}^\infty\) is convergent, and second, that the limit is \(x\text{.}\)
The definition above is one of the most important definitions in analysis, and it is necessary to understand it perfectly. The key point in the definition is that given any\(\epsilon > 0\text{,}\) we can find an \(M\text{.}\) The \(M\) can depend on \(\epsilon\text{,}\) so we only pick an \(M\) once we know \(\epsilon\text{.}\) Let us illustrate convergence on a few examples.
Example2.1.3.
The constant sequence \(1,1,1,1,\ldots\) converges to 1. For every \(\epsilon > 0\text{,}\) pick \(M = 1\text{.}\) That is, \(\sabs{x_n-x} = \sabs{1-1} < \epsilon\) for all \(n\text{.}\)
Example2.1.4.
Claim: The sequence \(\{ \nicefrac{1}{n} \}_{n=1}^\infty\) is convergent and
Proof: Given an \(\epsilon > 0\text{,}\) find an \(M \in \N\) such that \(0 < \nicefrac{1}{M} < \epsilon\) (Archimedean property at work). For all \(n \geq M\text{,}\)
The sequence \(\bigl\{ {(-1)}^n \bigr\}_{n=1}^\infty\) is divergent. Proof: If there were a limit \(x\text{,}\) then for \(\epsilon = \frac{1}{2}\) we expect an \(M\) that satisfies the definition. Suppose such an \(M\) exists. Then for an even \(n \geq M\text{,}\) we compute
The proof of this proposition exhibits a useful technique in analysis. Many proofs follow the same general scheme. We want to show a certain quantity is zero. We write the quantity using the triangle inequality as two quantities, and we estimate each one by arbitrarily small numbers.
Proof.
Suppose \(\{ x_n \}_{n=1}^\infty\) has limits \(x\) and \(y\text{.}\) Take an arbitrary \(\epsilon > 0\text{.}\) From the definition find an \(M_1\) such that for all \(n \geq M_1\text{,}\)\(\abs{x_n-x} < \nicefrac{\epsilon}{2}\text{.}\) Similarly, find an \(M_2\) such that for all \(n \geq M_2\text{,}\) we have \(\abs{x_n-y} < \nicefrac{\epsilon}{2}\text{.}\) Now take an \(n\) such that \(n \geq M_1\) and also \(n \geq M_2\text{,}\) and estimate
As \(\abs{y-x} < \epsilon\) for all \(\epsilon > 0\text{,}\) then \(\abs{y-x} = 0\) and \(y=x\text{.}\) Hence the limit (if it exists) is unique.
Proposition2.1.7.
A convergent sequence \(\{ x_n \}_{n=1}^\infty\) is bounded.
Proof.
Suppose \(\{ x_n \}_{n=1}^\infty\) converges to \(x\text{.}\) Thus there exists an \(M \in \N\) such that for all \(n \geq M\text{,}\) we have \(\abs{x_n - x} < 1\text{.}\) For \(n \geq M\text{,}\)
Therefore, \(\lim_{n\to\infty} \frac{n^2+1}{n^2+n} = 1\text{.}\) This example shows that sometimes to get what you want, you must throw away some information to get a simpler estimate.
Subsection2.1.1Monotone sequences
The simplest type of a sequence is a monotone sequence. Checking that a monotone sequence converges is as easy as checking that it is bounded. It is also easy to find the limit for a convergent monotone sequence, provided we can find the supremum or infimum of a countable set of numbers.
Definition2.1.9.
A sequence \(\{ x_n \}_{n=1}^\infty\) is monotone increasing if \(x_n \leq x_{n+1}\) for all \(n \in \N\text{.}\) A sequence \(\{ x_n \}_{n=1}^\infty\) is monotone decreasing if \(x_n \geq x_{n+1}\) for all \(n \in \N\text{.}\) If a sequence is either monotone increasing or monotone decreasing, we can simply say the sequence is monotone. 4
For example, \(\{ n \}_{n=1}^\infty\) is monotone increasing, \(\{ \nicefrac{1}{n} \}_{n=1}^\infty\) is monotone decreasing, the constant sequence \(\{ 1 \}_{n=1}^\infty\) is both monotone increasing and monotone decreasing, and \(\bigl\{ {(-1)}^n \bigr\}_{n=1}^\infty\) is not monotone. First few terms of a sample monotone increasing sequence are shown in Figure 2.2.
Figure2.2.First few terms of a monotone increasing sequence as a graph.
Theorem2.1.10.Monotone convergence theorem.
A monotone sequence \(\{ x_n \}_{n=1}^\infty\) is bounded if and only if it is convergent.
Furthermore, if \(\{ x_n \}_{n=1}^\infty\) is monotone increasing and bounded, then
Consider a monotone increasing sequence \(\{ x_n \}_{n=1}^\infty\text{.}\) Suppose first the sequence is bounded, that is, the set \(\{ x_n : n \in \N \}\) is bounded. Let
\begin{equation*}
x \coloneqq \sup \{ x_n : n \in \N \} .
\end{equation*}
Let \(\epsilon > 0\) be arbitrary. As \(x\) is the supremum, there must be at least one \(M \in \N\) such that \(x_{M} > x-\epsilon\text{.}\) As \(\{ x_n \}_{n=1}^\infty\) is monotone increasing, then it is easy to see (by induction) that \(x_n \geq x_{M}\) for all \(n \geq M\text{.}\) Hence for all \(n \geq M\text{,}\)
So \(\{ x_n \}_{n=1}^\infty\) converges to \(x\text{.}\) Therefore, a bounded monotone increasing sequence converges. For the other direction, we already proved that a convergent sequence is bounded.
The proof for monotone decreasing sequences is left as an exercise.
A monotone increasing sequence \(\{ x_n \}_{n=1}^\infty\) is always bounded from below since \(x_1 \leq x_2 \leq \cdots \leq x_n\) for any \(n\text{,}\) so \(x_1\) is a lower bound. So to see if a monotone increasing sequence is bounded, it is enough to check if it is bounded above. Similarly, a monotone decreasing sequence is always bounded from above, so it is enough to check whether it is bounded from below.
Example2.1.11.
Take the sequence \(\bigl\{ \frac{1}{\sqrt{n}} \bigr\}_{n=1}^\infty\text{.}\)
The sequence is bounded below as \(\frac{1}{\sqrt{n}} > 0\) for all \(n \in \N\text{.}\) Let us show that it is monotone decreasing. We start with \(\sqrt{n+1} \geq \sqrt{n}\) (why is that true?). From this inequality we obtain
We already know that the infimum is greater than or equal to 0, as 0 is a lower bound. Take a number \(b \geq 0\) such that \(b \leq \frac{1}{\sqrt{n}}\) for all \(n\text{.}\) We square both sides to obtain
\begin{equation*}
b^2 \leq \frac{1}{n} \qquad \text{for all } n \in \N.
\end{equation*}
We have seen before that this implies that \(b^2 \leq 0\) (a consequence of the Archimedean property). As \(b^2 \geq 0\) as well, we have \(b^2 = 0\) and so \(b = 0\text{.}\) Hence, \(b=0\) is the greatest lower bound, and \(\lim\limits_{n\to\infty} \frac{1}{\sqrt{n}} = 0\text{.}\)
Example2.1.12.
A word of caution: Showing that a monotone sequence is bounded in order to use Theorem 2.1.10 may be difficult. The sequence \(\{ 1 + \nicefrac{1}{2} + \cdots + \nicefrac{1}{n} \}_{n=1}^\infty\) is a monotone increasing sequence that grows slowly and in fact grows slower and slower as \(n\) gets larger. We will see, once we get to series, that this sequence has no upper bound and so does not converge. It is not at all obvious that this sequence has no upper bound.
A common example of where monotone sequences arise is the following proposition. The proof is left as an exercise.
Proposition2.1.13.
Let \(S \subset \R\) be a nonempty bounded set. Then there exist monotone sequences \(\{ x_n \}_{n=1}^\infty\) and \(\{ y_n \}_{n=1}^\infty\) such that \(x_n, y_n \in S\) and
For a sequence \(\{ x_n \}_{n=1}^\infty\text{,}\) the \(K\)-tail (where \(K \in \N\)), or just the tail, of \(\{ x_n \}_{n=1}^\infty\) is the sequence starting at \(K+1\text{,}\) usually written as
For example, the \(4\)-tail of \(\{ \nicefrac{1}{n} \}_{n=1}^\infty\) is \(\nicefrac{1}{5}, \nicefrac{1}{6}, \nicefrac{1}{7}, \nicefrac{1}{8},
\ldots\text{.}\) The \(0\)-tail of a sequence is the sequence itself. The convergence and the limit of a sequence only depends on its tail.
Proposition2.1.15.
Let \(\{ x_n \}_{n=1}^\infty\) be a sequence. Then the following statements are equivalent:
The sequence \(\{ x_n \}_{n=1}^\infty\) converges.
The \(K\)-tail \(\{ x_{n+K} \}_{n=1}^\infty\) converges for all \(K \in \N\text{.}\)
The \(K\)-tail \(\{ x_{n+K} \}_{n=1}^\infty\) converges for some \(K \in \N\text{.}\)
Furthermore, if any (and hence all) of the limits exist, then for all \(K \in \N\)
It is clear that ii implies iii. We will therefore show first that i implies ii, and then we will show that iii implies i. That is, In the process we will also show that the limits are equal.
We start with i implies ii. Suppose \(\{x_n \}_{n=1}^\infty\) converges to some \(x \in \R\text{.}\) Let \(K \in \N\) be arbitrary, and define \(y_n \coloneqq x_{n+K}\text{.}\) We wish to show that \(\{ y_n \}_{n=1}^\infty\) converges to \(x\text{.}\) Given an \(\epsilon > 0\text{,}\) there exists an \(M \in \N\) such that \(\abs{x-x_n} < \epsilon\) for all \(n \geq M\text{.}\) Note that \(n \geq M\) implies \(n+K \geq M\text{.}\) Therefore, for all \(n \geq M\text{,}\) we have
Consequently, \(\{ y_n \}_{n=1}^\infty\) converges to \(x\text{.}\)
Let us move to iii implies i. Let \(K \in \N\) be given, define \(y_n \coloneqq x_{n+K}\text{,}\) and suppose that \(\{ y_n \}_{n=1}^\infty\) converges to \(x \in \R\text{.}\) That is, given an \(\epsilon > 0\text{,}\) there exists an \(M' \in \N\) such that \(\abs{x-y_n} < \epsilon\) for all \(n \geq M'\text{.}\) Let \(M \coloneqq M'+K\text{.}\) Then \(n \geq M\) implies \(n-K \geq M'\text{.}\) Thus, whenever \(n \geq M\text{,}\) we have
Therefore, \(\{ x_n \}_{n=1}^\infty\) converges to \(x\text{.}\)
At the end of the day, the limit does not care about how the sequence begins, it only cares about the tail of the sequence. The beginning of the sequence may be arbitrary.
For example, the sequence defined by \(x_n \coloneqq \frac{n}{n^2+16}\) is decreasing if we start at \(n=4\) (it is increasing before). That is: \(\{ x_n \}_{n=1}^\infty =
\nicefrac{1}{17},
\nicefrac{1}{10},
\nicefrac{3}{25},
\nicefrac{1}{8},
\nicefrac{5}{41},
\nicefrac{3}{26},
\nicefrac{7}{65},
\nicefrac{1}{10},
\nicefrac{9}{97},
\nicefrac{5}{58},\ldots\text{,}\) and
If we throw away the first 3 terms and look at the 3-tail, it is decreasing. The proof is left as an exercise. Since the 3-tail is monotone and bounded below by zero, it is convergent, and therefore the sequence is convergent.
Subsection2.1.3Subsequences
It is useful to sometimes consider only some terms of a sequence. A subsequence of \(\{ x_n \}_{n=1}^\infty\) is a sequence that contains only some of the numbers from \(\{ x_n \}_{n=1}^\infty\) in the same order.
Definition2.1.16.
Let \(\{ x_n \}_{n=1}^\infty\) be a sequence. Let \(\{ n_i \}_{i=1}^\infty\) be a strictly increasing sequence of natural numbers, that is, \(n_i < n_{i+1}\) for all \(i \in \N\) (in other words \(n_1 < n_2 < n_3 < \cdots\)). The sequence
is called a subsequence of \(\{ x_n \}_{n=1}^\infty\text{.}\)
So the subsequence is the sequence \(x_{n_1},x_{n_2},x_{n_3},\ldots\text{.}\) Consider the sequence \(\{ \nicefrac{1}{n} \}_{n=1}^\infty\text{.}\) The sequence \(\{ \nicefrac{1}{3i} \}_{i=1}^\infty
= 1,\nicefrac{1}{3},\nicefrac{1}{6},\nicefrac{1}{9},\ldots\) is a subsequence. To see how these two sequences fit in the definition, take \(n_i \coloneqq 3i\text{,}\) that is, \(\{ n_i \}_{i=1}^\infty\) is the sequence \(3,6,9,12,\ldots\text{.}\) The numbers \(x_{n_i}\) in the subsequence must come from the original sequence. So \(1,0,\nicefrac{1}{3},0,
\nicefrac{1}{5},\ldots\) is not a subsequence of \(\{ \nicefrac{1}{n} \}_{n=1}^\infty\text{.}\) Similarly, order must be preserved. So the sequence \(1,\nicefrac{1}{3},\nicefrac{1}{2},\nicefrac{1}{5},\ldots\) is not a subsequence of \(\{ \nicefrac{1}{n} \}_{n=1}^\infty\text{.}\)
A tail of a sequence is one special type of a subsequence. For an arbitrary subsequence, we have the following proposition about convergence.
Proposition2.1.17.
If \(\{ x_n \}_{n=1}^\infty\) is a convergent sequence, then every subsequence \(\{ x_{n_i} \}_{i=1}^\infty\) is also convergent, and
It is not hard to prove (do it!) by induction that \(n_i \geq i\) for all \(i \in \N\text{.}\) Hence \(i \geq M\) implies \(n_i \geq M\text{.}\) Thus, for all \(i \geq M\text{,}\)
Existence of a convergent subsequence does not imply convergence of the sequence itself. Take the sequence \(0,1,0,1,0,1,\ldots\text{.}\) That is, \(x_n = 0\) if \(n\) is odd, and \(x_n = 1\) if \(n\) is even. The sequence \(\{ x_n \}_{n=1}^\infty\) is divergent; however, the subsequence \(\{ x_{2i} \}_{i=1}^\infty\) converges to 1 and the subsequence \(\{ x_{2i+1} \}_{i=1}^\infty\) converges to 0. Compare Proposition 2.3.7.
Subsection2.1.4Exercises
In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the sequence is divergent.
Exercise2.1.1.
Is the sequence \(\{ 3n \}_{n=1}^\infty\) bounded? Prove or disprove.
Exercise2.1.2.
Is the sequence \(\{ n \}_{n=1}^\infty\) convergent? If so, what is the limit?
Exercise2.1.3.
Is the sequence \(\left\{ \dfrac{{(-1)}^n}{2n} \right\}_{n=1}^\infty\) convergent? If so, what is the limit?
Exercise2.1.4.
Is the sequence \(\{ 2^{-n} \}_{n=1}^\infty\) convergent? If so, what is the limit?
Exercise2.1.5.
Is the sequence \(\left\{ \dfrac{n}{n+1} \right\}_{n=1}^\infty\) convergent? If so, what is the limit?
Exercise2.1.6.
Is the sequence \(\left\{ \dfrac{n}{n^2+1} \right\}_{n=1}^\infty\) convergent? If so, what is the limit?
Exercise2.1.7.
Let \(\{ x_n \}_{n=1}^\infty\) be a sequence.
Show that \(\lim\limits_{n\to\infty} x_n = 0\) (that is, the limit exists and is zero) if and only if \(\lim\limits_{n\to\infty} \abs{x_n} = 0\text{.}\)
Find an example such that \(\{ \abs{x_n} \}_{n=1}^\infty\) converges and \(\{ x_n \}_{n=1}^\infty\) diverges.
Exercise2.1.8.
Is the sequence \(\left\{ \dfrac{2^n}{n!} \right\}_{n=1}^\infty\) convergent? If so, what is the limit?
Exercise2.1.9.
Show that the sequence \(\left\{ \dfrac{1}{\sqrt[3]{n}} \right\}_{n=1}^\infty\) is monotone and bounded. Then use Theorem 2.1.10 to find the limit.
Exercise2.1.10.
Show that the sequence \(\left\{ \dfrac{n+1}{n} \right\}_{n=1}^\infty\) is monotone and bounded. Then use Theorem 2.1.10 to find the limit.
Exercise2.1.11.
Finish the proof of Theorem 2.1.10 for monotone decreasing sequences.
Show that \(x_n = x_k\) for all \(n \geq k\text{.}\)
Exercise2.1.14.
Find a convergent subsequence of the sequence \(\bigl\{ {(-1)}^n \bigr\}_{n=1}^\infty\text{.}\)
Exercise2.1.15.
Let \(\{x_n\}_{n=1}^\infty\) be a sequence defined by
\begin{equation*}
x_n \coloneqq
\begin{cases}
n & \text{if } n \text{ is odd} , \\
\nicefrac{1}{n} & \text{if } n \text{ is even} .
\end{cases}
\end{equation*}
Is the sequence bounded? (prove or disprove)
Is there a convergent subsequence? If so, find it.
Exercise2.1.16.
Let \(\{ x_n \}_{n=1}^\infty\) be a sequence. Suppose there are two convergent subsequences \(\{ x_{n_i} \}_{i=1}^\infty\) and \(\{ x_{m_i} \}_{i=1}^\infty\text{.}\) Suppose
where \(a \not= b\text{.}\) Prove that \(\{ x_n \}_{n=1}^\infty\) is not convergent, without using Proposition 2.1.17.
Exercise2.1.17.
(Tricky) Find a sequence \(\{ x_n \}_{n=1}^\infty\) such that for every \(y \in \R\text{,}\) there exists a subsequence \(\{ x_{n_i} \}_{i=1}^\infty\) converging to \(y\text{.}\)
Exercise2.1.18.
(Easy) Let \(\{ x_n \}_{n=1}^\infty\) be a sequence and \(x \in \R\text{.}\) Suppose for every \(\epsilon > 0\text{,}\) there is an \(M\) such that \(\abs{x_n-x} \leq \epsilon\) for all \(n \geq M\text{.}\) Show that \(\lim\limits_{n\to\infty} x_n = x\text{.}\)
Exercise2.1.19.
(Easy) Let \(\{ x_n \}_{n=1}^\infty\) be a sequence and \(x \in \R\) such that there exists a \(k \in \N\) such that for all \(n \geq k\text{,}\)\(x_n = x\text{.}\) Prove that \(\{ x_n \}_{n=1}^\infty\) converges to \(x\text{.}\)
Exercise2.1.20.
Let \(\{ x_n \}_{n=1}^\infty\) be a sequence and define a sequence \(\{ y_n \}_{n=1}^\infty\) by \(y_{2k} \coloneqq x_{k^2}\) and \(y_{2k-1} \coloneqq x_k\) for all \(k \in \N\text{.}\) Prove that \(\{ x_n \}_{n=1}^\infty\) converges if and only if \(\{ y_n \}_{n=1}^\infty\) converges. Furthermore, prove that if they converge, then \(\lim\limits_{n\to\infty} x_n = \lim\limits_{n\to\infty} y_n\text{.}\)
Exercise2.1.21.
Show that the 3-tail of the sequence defined by \(x_n \coloneqq \frac{n}{n^2+16}\) is monotone decreasing. Hint: Suppose \(n \geq m \geq 4\) and consider the numerator of the expression \(x_n-x_m\text{.}\)
Exercise2.1.22.
Suppose that \(\{ x_n \}_{n=1}^\infty\) is a sequence such that the subsequences \(\{ x_{2n} \}_{n=1}^\infty\text{,}\)\(\{ x_{2n-1} \}_{n=1}^\infty\text{,}\) and \(\{ x_{3n} \}_{n=1}^\infty\) all converge. Show that \(\{ x_n \}_{n=1}^\infty\) is convergent.
Exercise2.1.23.
Suppose that \(\{ x_n \}_{n=1}^\infty\) is a monotone increasing sequence that has a convergent subsequence. Show that \(\{ x_n \}_{n=1}^\infty\) is convergent. Note: So Proposition 2.1.17 is an “if and only if” for monotone sequences.
It is common to use \(\{ x_n \}\) or \(\{ x_n \}_n\) for brevity.
[BS] use \((x_n)_{n=1}^\infty\) to denote a sequence instead of \(\{ x_n \}_{n=1}^\infty\text{,}\) which is what [R2] uses. Both are common.
In text, this may get rendered as \(\lim_{n\to\infty} x_n\text{.}\)
Some authors use the word monotonic.
For a higher quality printout use the PDF versions: https://www.jirka.org/ra/realanal.pdf or https://www.jirka.org/ra/realanal2.pdf
In the process we will also show that the limits are equal.
