Intuitively this makes sense to me. If something converges, then that means the terms eventually get small enough so that they are infinitely small. I just can't put it into formal terms through a proof. Any ideas?
This is number 2 of the problem set for real analysis problems:
Prove that if
converges uniformly, then (g_n) converges uniformly to zero.
If converges uniformly then it is uniformly Cauchy.
Let then there is so that if
Then, for .
Thus, for example if then, .
Thus, we see that uniformly on .