Mathematics — Real Analysis · pointwise-vs-uniform-convergence
Problem
Consider a sequence of functions fn:[0,1]→R so that there is a sequence of distinct numbers xn∈[0,1] such that for all n,
fn(xn)=1.
Prove or disprove the following statements:
(a)) True or false: There exists {fn}n=1∞ as above that converges pointwise to 0.
(b)) True or false: There exists {fn}n=1∞ as above that converges uniformly to 0.