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Here is the theorem:

Let $\{s_n \}$ and $\{t_n \}$ be sequences of real numbers. If $s_n\leq t_n$ for $n\geq N$, where $N$ is fixed, then $\lim_{n\to\infty}\inf s_n\leq\lim_{n\to\infty}\inf t_n$

$\lim_{n\to\infty}\sup s_n\leq\lim_{n\to\infty}\sup t_n$

Here's my attempted proof:

By definition: $\lim_{n\to\infty}t_n\leq\lim_{n\to\infty}\sup t_n$

This means there exists a $M$ such that, for each $n\geq M$

$t_n\leq\lim_{n\to\infty}\sup t_n=t^*$

If we pick $n\geq\max\{N,M\}$ then we have

$t^*\geq t_n\geq s_n$

Therefore for each subsequence $s_{n(k)}$ we must have (for each $n(k)\geq\max\{N,M\}$)

$t^*\geq s_{n(k)}$

Therefore the set of all subsequential limit of $\{s_n \}$ is bounded above by $t^*$ which means that $s^*$ is not bigger than $t^*$ itself ($t^*\geq s^*$)

Is this correct?

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    $\begingroup$ Your attempted proof fails from the very beginning: $\lim_{n\to\infty}t_n$ does not need to exist. $\endgroup$ – Andrés E. Caicedo Sep 27 '17 at 19:11
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    $\begingroup$ You cannot assume the sequence converges. $\endgroup$ – Clayton Sep 27 '17 at 19:12
  • $\begingroup$ You're both right, i assumed the sequence converged. Is there a way to fix my proof so it would work somehow? I don't know, maybe proving the theorem in two cases: one when $\{t_n \}$ does converge and one when it does not? $\endgroup$ – Nikolaj Di Rondò Sep 27 '17 at 19:23
  • $\begingroup$ See also this question and this question. $\endgroup$ – Arnaud D. Dec 5 at 13:37
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Note that: $$\lim_n \sup x_n=\inf_n \sup\{x_k:k \geq n\}$$ $$\lim_n \inf x_n =\sup_n \inf \{x_k:k \geq n\}$$

Thus $$ \sup\{s_k:k \geq N\} \leq \sup\{t_k:k \geq N\}$$ so $$\inf_n \sup\{x_k:k \geq n\} \leq \inf_n \sup\{t_k:k \geq n\}$$

I leave the other case for you.

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