Okay, this problem has been kicking my butt for several hours now.

Consider the positive sequences and with

Prove or disprove: If converges to , then

converges to .

I attempted to prove this as follows:

Since

Therefore

Thus

Did I even remotely do that right? I feel like I should be using the

triangle inequality to prove this, but I haven't been able to produce

a useful result with it. The best I have is:

I could probably get a more useful result if , but I can't find a single effective way to prove that