Theorems of convergence for sequences?

[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!

 

[fresh_42]

Since $u$ doesn't occur in your second definition, there is something wrong. And what if $u(0)\neq 0\,?$ I guess you ask why the $\varepsilon-\delta-$definition of continuity and the sequence definition of continuity is the same.
https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Metric_Space_Continuity_at_Point