# Theorems of convergence for sequences?

1. Oct 18, 2005

### cappygal

For a sequence a_1, a_2, ... in R^n to be convergent there are (at least) 2 theorems, as follows:

if for all epsilon>0 there exists an M such that when m>M, then |a_m-a|<epsilon

and also:
If u(epsilon) is a function such that u(epsilon)-->0 as epsilon-->0, then
the sequence is convergent if

for all epsilon>0 there exists an M such that when m>M, then |a_m-a|<epsilon

~~~~~~~~~~~~
I can understand this intuitively, because u(epsilon) and epsilon behave similarly .. but how I prove that they are equivalent? Do I simply take the limit of both definitions, and then set them equal? help!