(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given the sequence {a_n} converges to A and {b_n} converges to B, and a_n <_ b_n for all n>_ n*, prove A <_ B

2. Relevant equations

x+ epsilon < y for every positive real epsilon, implies x <_ y

A - B = (A - a_n) + (b_n - B) + (a_n - b_n)

3. The attempt at a solution

I want to show A - B is not positive, I know a_n - b_n is not positive by hypothesis, and A - a_n and b_n - B are bounded by (-e/2, e/2) where e is any postive number, hence their sum is bounded by (-e,e). But since their sum is not bounded by zero, I can't figure out how to apply (x + e) - y negative implies (x - y) non positive.

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# Prove the limit theorem: a_n < b_n -> A-B

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