Define $ G_ {1}: = {f_ {n} (x): = sum limits_ {i = 1} ^ {n} frac {x ^ {i}} {i!} | N in mathbb N } subseteq C ([0,1]$

The background to this question is that I want to show that $ G_ {1} $ It is relatively compact. It is clear that I need to use Arzela-Ascoli, but to do that, first I have to demonstrate the continuity of equity, and this is where I am fighting.

I know that $ (f_ {n}) _ {n} $ converges uniformly in $[0,1]$ For something $ f $ Y $ f_ {n} (x) leq exp (x) $. From this I know that by any $ epsilon> 0 $ exists $ N en mathbb N $ so that $ vert vert f_ {n} -f vert vert _ { infty} < epsilon $ for all $ n geq N $, where $ vert vert f_ {n} -f vert vert _ { infty} = sup limits_ {x in [0,1]} vert f_ {n} (x) -f (x) vert $ But how does this help me to show continuity and much less to continuity?