Prove that fn is converge uniformly to 0

Question

In the first section of this problem which about f and it's solution, I try to cheak it but I confused about who to prove uniformity to this given example ?

Answer 1

To prove uniformity in (a) you need to show $|f_m(x)-f_n(x)|<\epsilon$ for all $m,n>N$, which amounts to showing that $|m^{-1}-n^{-1}|<\epsilon$, which follows when $N=1/\epsilon$. Part (b) is a standard argument based on non-uniform 'convergence to a delta-function'. Both parts relate directly to the Dominated Convergence Theorem and the necessity of its assumptions.