How should I prove a uniformly continuous sequence of real-valued continuous function is equicontinuous? how should I handle case when $n \leq N$?
This is what I have so far.
Given $\epsilon>0$, there exists N s.t. for all $n\geq N$ we have $|f_n-f|<\epsilon$.
Whenever $\rho(x,y)<\epsilon$, $|f_n(x)-f_n(y)|\leq|f_n(x)-f(x)|+|f(x)-f(y)|+|f(y)-f_(y)|$.
However, how should I handle case when $n \leq N$?