# Proof that for two converging sequences and two corresponding unbound sets - the limit is equal.

Ok so I know I've seen this before phrased differently but I can't quite put my finger on the solution. The question goes as follows:

Let $$a_n, b_n$$ be two converging sequences.

Also, let $$\{n: n\in\Bbb{N}, a_n \leq b_n\}$$ and $$\{n: n\in\Bbb{N}, b_n \leq a_n\}$$ be two unbounded sets.

Prove that $$\lim\limits_{n\to \infty}a_n = \lim\limits_{n\to \infty}b_n$$

I'm completely drawing a blank. I tried several approaches I know, such as Cantor's intersection theorem (which doesn't seem to fit) or other properties of converging sequences I've learned, but the question - as posed - doesn't provide enough detail (that I can see) to use any of those as far as I can tell.

Can anyone shed some light on this for me?

Thanks!

Edit (possible solution?) After some page-turning I found a similar proof in my professor's slides and adapted it to this question. Let me know if you think it's valid:

First we'll show that if $$a_n \leq b_n$$ for every $$n$$, then $$\lim\limits_{n\to \infty}a_n \leq \lim\limits_{n\to \infty}b_n$$:

Assuming by contradiction that if if $$a_n \leq b_n$$ for every $$n$$, then $$\lim\limits_{n\to \infty}a_n > \lim\limits_{n\to \infty}b_n$$, there then exists an $$n$$ in $$a_n$$ for which $$a_n>b_n$$, in contradiction to the fact that $$a_n \leq b_n$$ for every $$n$$, and therefor if $$a_n \leq b_n$$ for every $$n$$, then $$\lim\limits_{n\to \infty}a_n \leq \lim\limits_{n\to \infty}b_n$$.

We use the same process to prove the opposite direction, and since we have two infinite sets of indices for which $$a_n\geq b_n$$ and $$b_n \geq a_n$$, therefor $$\lim\limits_{n\to \infty}a_n \leq \lim\limits_{n\to \infty}b_n$$ and $$\lim\limits_{n\to \infty}b_n \leq \lim\limits_{n\to \infty}a_n$$ - which means $$\lim\limits_{n\to \infty}a_n = \lim\limits_{n\to \infty}b_n$$ - Q.E.D.

• What do you know about the subsequences of a convergent sequence? – Daniel Fischer Apr 5 '15 at 8:47
• Technically we haven't covered that topic yet, but I know from previous courses that all sub-sequences of a convergent sequence have the same sub-limit, except I can't use that because we haven't even touched the term 'subsequence' yet in class. – Elad Avron Apr 5 '15 at 8:49
• It just occured to me that the mentioned sets are not of members in the sequences but of INDICES. Could this help me in any way? I can't seem to think how. – Elad Avron Apr 5 '15 at 8:50
• But if you could use subsequences, would you see which subsequences you could take to reach the desired conclusion? If so, you can translate that idea into a proof not mentioning subsequences. – Daniel Fischer Apr 5 '15 at 8:51
• As you emphasised above: indices of subsequences. For subsets of $\mathbb{N}$, "unbounded" and "infinite" are equivalent, so if one of the two sets were bounded, we wouldn't have the corresponding subsequence. – Daniel Fischer Apr 5 '15 at 8:57

Let the limits be $a$ and $b.$
Fix $\epsilon > 0.$ There exist $N_{\epsilon}$ such that for $n>N_{\epsilon}$ $$|a_n- a | < \epsilon, \; |b_n-b| < \epsilon.$$ or alternatively $$a-\epsilon < a_n < a+\epsilon, \; b-\epsilon < b_n < b+\epsilon.$$ There is some such $n$ with $a_n < b_n.$ So $$a-\epsilon < b + \epsilon.$$ Similarly, $$b-\epsilon < a + \epsilon.$$ So $$|a-b | < 2\epsilon.$$ But $\epsilon$ was arbitrary, so $a=b.$