Answer to Question #241910 in Functional Analysis for smi

Question

If x_n∈C[a,b] and x_n→x∈C[a,b]. Show that (x_n) is pointwise convergent on [a,b], that is, (x_n(t)) converges for every t∈C[a,b].

Answer 1

A sequence of functions $x_1,x_2,x_3,...,x_n,...$ is said to be converges pointwise if and only if

x_n→x for all  t∈C[a,b].

So, sequence (x_n) is pointwise convergent on [a,b].