Suppose ${f_n}$ is a sequence of nonnegative continuous functions defined on $[0, 1]$ and suppose $lim_{n→∞} f_n(x) = 0$ pointwise on $[0, 1]$.
Prove that, $∀\varepsilon> 0$, there exist $δ > 0$, $N ∈ Z +$, and points $x_1, · · · , x_N$ and $n_1, · · · , n_N$ such that $$∪_{k=1}^N[x_k − δ, x_k + δ] ⊃ [0, 1]$$, and $$0 ≤ f_{n_k}(x) <\varepsilon , ∀x ∈ [x_k − δ, x_k + δ] ∩ [0, 1], ∀1 ≤ k ≤ N$$.
Since I know that $f_n$ converges pointwise, can it be proved inductively by applying open ball/neighbourhood definition that comes with pointwise convergence and continuity?