Proof of a theorem with upper/lower limits.

Question

Theorem: If $s_n \le t_n$ for all $n$ greater than a fixed integer $N$, then $$\lim_{n \to \infty} \inf s_n \le \lim_{n \to \infty} \inf t_n$$

I would like to prove this and it would be nice if someone could check my work.

Proof: Letting $$\lim_{n \to \infty} \inf s_n = s_*$$ and $$\lim_{n \to \infty} \inf t_n = t_*$$ assume that $s_* > t_* $ while there exists $N$ such that for all $n$ >$N$, $s_n \le t_n$. Then there must be an integer $M$(particularly larger than $N$) such that if $m > M$, $t_m < s_*$. This contradicts the fact that for all $n > N$, $s_n \le t_n$.

score 1 · Accepted Answer · answered May 17 '13 at 06:54

1

It is right. Another nice proof is this:

For any $N \le n$, $s_n\le t_n$, and hence $\inf \{s_n: N\le n\} \le \inf \{t_n: N\le n\}$. Then we can conclude that $$\lim_{n \to \infty} \inf s_n \le \lim_{n \to \infty} \inf t_n.$$ This complete the proof.

answered May 17 '13 at 06:54

Paul

20,553

Thanks ! I'm still not quite sure how subsequences and stuff works, so knowing that I did one problem right makes me feel better. – hyg17 May 17 '13 at 07:12
@hyg17: You are welcome. – Paul May 17 '13 at 07:15

Proof of a theorem with upper/lower limits.

1 Answers1