I'm having trouble with the solution to following problem:
a) Suppose that $\{f_n\}$ is a sequence of continuous functions on [a,b] which approaches $0$ pointwise. Suppose moreover that we have $f_n(x)\geq f_{n+1}\geq 0$ for al n and all x in [a,b]. Prove that ${f_n}$ actually approaches $0$ uniformly on [a,b].
c) Does it fail if f isn't continuous? How about if $[a,b]$ is replaced by the open interval $(a,b)$.
Solution I read:
1) As $f_n$ converges pointwise to $f$ and it's continuous, for any $\varepsilon>0$ we can find a neighborhood around x such that $0\leq f_n(x)<\epsilon$ for all x in this interval for an n greater than a certain integer $n_x$.
2) Trying to prove that there is a finite set that is a subcover of the interval. Then, as we have such finite set we can consider a $max$ of the individuals $n_x$
3) Taking the previously considered $max$, we can just conclude that $f_n<\varepsilon$ for all x in the interval, as we have considered a subcover, thus arriving at the necessary condition. (When does pointwise convergence on $[a,b]$ imply uniform convergence on $[a,b]$)
My Problem:
I don't understand fully covers and subcovers, though I've read the definitions of each the lack of previous exposure to them makes me desire a more in-depth explanation of that specific part or an alternative solution that doesn't rely on them. What bothers me the most is that passage from a cover to a finite subcover. This makes me uncertain when solving item c as well. If possible, a solution using Bolzano-Weierstrass would be greatly appreciated.