Show that the $\lim \inf a_n = \lim \sup a_n$ if and only if $\lim a_n$ exists.
Attempt at proof.
($ \Rightarrow $) Assume $\lim \inf a_n = \lim \sup a_n= L$. Then we have $|\sup a_n -L| \lt \epsilon$ and $|\inf a_n – L| \lt \epsilon$ and thus $$L-\epsilon \lt \sup a_n \lt \epsilon +L $$
and
$$L -\epsilon \lt \inf a_n \lt L + \epsilon$$
and since $\inf a_n \le a_n\le \sup a_n$ it follows that $L- \epsilon \lt a_n\lt L + \epsilon$ i.e. $\lim a_n =L$
($\Leftarrow$)
Let $\lim a_n = L$. Assume that $\lim \inf a_n \neq \lim \sup a_n$ Choose $N_0$ such that $n \ge N_0$ implies $|a_n- L| \lt \epsilon = \frac{\sup a_n- \inf a_n}{2}$
Then we see that $$|\sup a_n – a_n| \le |a_n- L| \lt \frac{\sup a_n- \inf a_n}{2}$$
and
$$|\inf a_n – a_n| \le |a_n- L| \lt \frac{\sup a_n- \inf a_n}{2}$$
By the Triangle Inequality we have $$|\sup a_n- \inf a_n| \lt |\sup a_n – a_n| + |a_n – \inf a_n| \lt 2\frac {\sup a_n- \inf a_n}{2}$$
which is a contradiction.
Can you please provide the correct proof. I'm pretty sure the second direction is incorrect. I am having a hard time showing that $|\sup a_n – a_n| \le |a_n- L|$ and that $|\inf a_n – a_n| \le |a_n- L|$ could be true. Thanks in advance.
Best Answer
You have issues with both the directions and you need to understand the meaning of $\liminf, \limsup$ properly. The problem in first part is the inequality $\liminf a_n\leq a_n\leq \limsup a_n$ which is false (check using the sequence $a_n=(-1)^n(1+n^{-1})$).
And you have already mentioned the problem with second part. So let us now see how to do this properly. Assume $\liminf a_n=\limsup a_n=L$ and let $\epsilon >0$ be arbitrary. By definition of $\limsup, \liminf $ there exist positive integers $n_1,n_2$ such that $$L-\epsilon=\liminf a_n-\epsilon <a_n, \forall n\geq n_1\\ a_n<\limsup a_n+\epsilon=L+\epsilon, \forall n\geq n_2$$ and therefore if $n\geq n_0=\max(n_1,n_2)$ then we have $$L-\epsilon<a_n<L+\epsilon, \forall n\geq n_0$$ and this means that $\lim_{n\to\infty} a_n=L$.
Next the second direction is easy. Assume that $\lim_{n\to\infty} a_n=L$ and then we show that $\limsup a_n=\liminf a_n=L$. Let $\epsilon >0$ be arbitrary. Then there is a positive integer $n_0$ such that $$L-\epsilon<a_n<L+\epsilon,n\geq n_0$$ By definition of $\liminf, \limsup$ the conclusion follows from the above inequality.