Sequence converges iff $\limsup = \liminf$
I want to prove that a sequence of real numbers $\{s_n\}$ converges to $s$ if and only if $\limsup_{n \to \infty} s_n = \liminf_{n \to \infty} s_n = s$.
Here are my definitions:
For any sequence of real numbers $\{s_n\}$, let $E$ be the set of all subsequential limits of $\{s_n\}$, including possibly $+\infty$ and/or $-\infty$ if any subsequence of $\{s_n\}$ diverges to infinity. Then $\limsup_{n \to \infty} s_n = \sup E$, and $\liminf_{n \to \infty} s_n = \inf E$.
I know the theorem that a sequence converges to a point if and only if every one of its subsequences converge to that same point, so one direction of this proof is easy:
If $\{s_n\}$ converges to some point $s \in \mathbb{R}$, then every subsequence of $\{s_n\}$ converges to $s$. So the set $E$ of every subsequential limit of $\{s_n\}$ consists of the single point $s$, so $$\limsup_{n \to \infty} s_n = \sup \{s\} = s = \inf \{s\} = \liminf_{n \to \infty} s_n$$
But the other direction seems more tricky...
If $\limsup_{n \to \infty} s_n = \liminf_{n \to \infty} s_n = s$, then every convergent subsequence converges to the same point $s$. Also, there can be no subsequences which diverge to infinity (otherwise $\limsup_{n \to \infty} s_n$ would be $+\infty$, or $\liminf_{n \to \infty} s_n$ would be $-\infty$).
But can't there be subsequences which diverge otherwise? And wouldn't that throw off the convergence of $\{s_n\}$?
EDIT:
I'd also be willing to accept a solution which makes use of the "Pinching Theorem" (if $a_n \leq s_n \leq b_n$ for every $n \in \mathbb{N}$, and if $a_n \to s$ and $b_n \to s$, then $s_n \to s$).
Solution 1:
Useful facts that you should verify:
Any unbounded sequence has a subsequence diverging to $\infty$ or to $-\infty$.
Any bounded sequence has a convergent subsequence.
You correctly point out that the hypothesis that $\limsup_{n \to \infty} s_n$ and $\liminf_{n \to \infty} s_n$ are both finite implies that $(s_n)_{n=1}^{\infty}$ has no subsequences that diverge to infinity. But (1) implies that more is true: the sequence $(s_n)_{n=1}^{\infty}$ must be bounded.
Fix any $\epsilon > 0$.
-
There cannot be infinitely many $n$ for which $s_n \geq s + \epsilon$, because you could select out of them a subsequence $y_k = s_{n_k}$ satisfying $y_k \geq s + \epsilon$ for all $k$.
- Since the sequence $(s_n)_{n=1}^{\infty}$ is bounded, so is the sequence $(y_k)_{k=1}^{\infty}$. So by (2) it has a subsequence convergent to some limit $L$; and by basic facts about limits, since $y_k \geq s + \epsilon$ for all $k$, one must have $L \geq s + \epsilon$.
- But $L$ is clearly also a subsequential limit of $s$, allowing us to deduce that $\limsup_{n \to \infty} s_n \geq s + \epsilon$, a contradiction.
Similarly, there cannot be infinitely many $n$ for which $s_n \leq s - \epsilon$.
So there is a positive integer $N$ with the property that whenever $n \geq N$ one has $$ s - \epsilon < s_n < s + \epsilon, $$ and since $\epsilon > 0$ was arbitrary, $(s_n)_{n=1}^{\infty}$ converges to $s$.
Solution 2:
Another possible solution (for bounded case only), just use these facts:
$\inf A=\sup A \quad\Leftrightarrow\quad A=\{\text{single element}\}.$ (A bounded and nonempty) -Proof
$(x_n)\text{ bounded and every convergent subsequence of } x_n\text{ converges to } a \quad\Rightarrow \quad\lim(x_n)=x$$ -Proof
Solution 3:
Here's my proof based on more basic observations.
Setup: For a sequence $(s_n), n \in \mathbb{N}$, if we define $s^+_n = \sup_{j\geq n}(s_j)$ and $s^-_n = \inf_{j\geq n}(s_j)$, then we can equivalently define $\lim \sup_{n\to \infty} s_n := \inf_n s^+_n$ and $\lim \inf_{n\to \infty} s_n := \sup_n s^-_n$ (this is actually how my analysis book by Terence Tao defined them; also see this discussion). I'll use $L^+=\lim \sup_{n\to \infty} s_n$ and $L^-=\lim \inf_{n\to \infty} s_n$ for convenience.
Now on to the proof:
- If a sequence $(s_n)$ converges to $s\in\mathbb{R}$, then $\lim \sup_{n\to\infty} s_n=\lim \inf_{n\to\infty} s_n =s$.
Proof: Let's just prove $\lim \sup_{n\to\infty} s_n=s$; then $\lim \inf_{n\to\infty} s_n =s$ can be proved similarly. Since $(s_n)$ converges, $(s_n^+)$ must also converge (because it's also bounded, and is also monotonically decreasing), then it's not hard to show $\lim \sup_{n\to\infty} s_n = \inf_n s^+_n = \lim_{n\to \infty} s^+_n = L^+ $. Note that $\forall n \in \mathbb{N}, s_n \leq s^+_n$, therefore $\lim_{n\to\infty} s_n \leq \lim_{n\to \infty} s^+_n $, i.e., $s \leq L^+$. Now we just need to show it's not the case that $s < L^+$. For contradiction, suppose $s < L^+$. Since $(s_n)$ converges to $s$, there exists $N$ s.t. $\forall k\geq N$, $|s_n-s| \leq \frac{2(L^+-s)}{3}$, and thus $\forall k\geq N$, $s_k \leq s+\frac{2(L^+-s)}{3} <L^+ - \frac{L^+-s}{3}$, which means $s^+_N = \sup_{k\geq N} (s_k) \leq L^+- \frac{L^+-s}{3}$. But $L^+:= \inf_n s^+_n$, so $L^+ \leq s^+_N$, then $L^+ \leq L^+- \frac{L^+-s}{3}$, a contradiction.
- If $\lim \sup_{n\to\infty} s_n=\lim \inf_{n\to\infty} s_n =s\in\mathbb{R}$, then the sequence $(s_n)$ converges to $s$.
Proof: Again we observe that we also have $L^+=\lim_{n\to\infty}(s^+_n)=s$ and $L^-=\lim_{n\to\infty}(s^-_n)=s$. Let arbitrary $\epsilon > 0$. Then $\exists N_1 \in \mathbb{N}$, .s.t. $\forall k \geq N_1, s^+_k -s \leq \epsilon$; similarly $\exists N_2 \in \mathbb{N}$, .s.t. $\forall k \geq N_2, s-s^-_k \leq \epsilon$. Let $N=\max(N_1,N_2$). Then $\forall k \geq N$, $s-\epsilon \leq s^-_k \leq s_k \leq s^+_k \leq s+\epsilon$, hence $|s_k - s| \leq \epsilon$.