
power series convergence
i was reading a proof on :
given that the power series converges on (R,R) then it is continuous on the open interval(R,R).
i know that i have to split it up into using triangular inequality where for all ε>0
let
δ'>o st l f(x)  f_m(x) l < ε/3
and
δ'' >0 st l f_m(x)  f_m(c) l < ε/3 ( continous at c).
im wondering, does δ' and δ'' depend on ε or x?

This is a relatively difficult theorem. Its proof pretty much goes along the lines of the proof of Abel's Theorem.
It's not true for series of functions in general; for example, trigonometric series may be discontinuous. This is something special about power series.

does δ depend on ε in the proof?

It very well can, yes, as for any proof that a function is continuous at a point.