Paper 2016 Solution

2.3 For each $n \ge 1$ , let fn be a monotonic increasing real valued function on $[0, 1]$ such that the sequence of functions { $f_{n}$ } converges pointwise to the function $f \equiv 0$ . Pick out the true statements from the following:
a. $f_{n}$ converges to $f$ uniformly.
b. If the functions $f_{n}$ are also non-negative, then $f_{n}$ must be continuous for sufficiently large $n$ .

Answer For each $n \ge 1$ , let fn be a monotonic increasing real valued function on $[0, 1]$ such that the sequence of functions ${fn}$ converges pointwise to the function f ≡ 0. Pick out the true statements from the following:
a. fn converges to f uniformly.
b. If the functions fn are also non-negative, then fn must be continuous for sufficiently large n.