I am having trouble coming up with a proof for this problem about a sequence of continuous functions that converges (pointwise) to 0 on [0,1].
https://i.stack.imgur.com/FE7Wr.png
Proving the first part of the problem is pretty simple since [0,1] is compact therefore we can convert it for any delta >0 we can cover it with a finite collection of closed balls centered at points in [0,1] with radius=delta.
What I can't figure out is which delta to take and how to then show that {fn} is uniformly convergent to 0 in each intersection with the closed balls.
Help would be greatly appreciated.