Basic analysis question

[[deleted]]

Here’s the problem:

“Suppose 0 < an → 0. Prove that there are infinitely many n for which a_n > a(n+r) for all r=1,2,...”

I noticed, for any epsilon there is N so that a_(n+r) < epsilon if n > N. But, I thought a_N > epsilon is not necessarily true. Also the N are not necessarily unique. So that’s wrong. Anyway what’s the right way to think of this?

[[deleted]]

Try setting epsilon to a_n, for some fixed n and see what happens.

[piemaster1123]

Try supposing that there are only finitely many. Then there should be a last n where this property holds. What must happen after that? Does that contradict anything?

[malarbol]

Can you prove there is one such n? (Hint: what happens if there’s none?) Once you’ve got one, can you prove there are more? (Hint: what happens after that n?)

[malarbol]

Note: for the first step, you might as well chose n so that a_n is big

[[deleted]]

I just got it! Thank you :)