A basic problem on sup and uniform convergence

Question

Suppose $\{f_n\}$ converges uniformly to $f$. Now, can we find an $N$ such that for all $n \geq N$ $\sup (f_n - f) < \epsilon$ for all $n \geq N$. How to choose such an $N$ ?

Answer 1

Since $f_n \rightarrow f$ uniformly, we know that for some $N'$ we have that for all $x$ in the domain and $n >N'$, $$|f_n(x) - f(x)| < \epsilon$$ $\epsilon$ is an upper bound for $f_n -f$, so the least upper bound must also be less than $\epsilon$. Thus, whatever $N$ provides uniform convergence at level $\epsilon$ will work for the least upper bound. $$N = N'$$