Hello everyone in the math community,

Here's a problem I hacked at for a bit and can't quite seem to find an adequate solution for (regardless of my approach, the differences I set up in absolute value never quite seem to cancel properly).

Suppose

is a Cauchy sequence (the usual properties; for all epsilon, there is an index beyond which the distance between two elements is less than epsilon). In addition,

has the property that for all epsilon, there exists an index k > 1 / epsilon, such that:

epsilon.

Prove that

converges to 0.

Now I understand the gist of the image; arbitrarily close to 0 we will find SOME term of the sequence close to 0, and the Cauchy property will force the rest of the sequence to lie near 0. However, the actual mechanics of the proof aren't quite clicking for me. Help?