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Moderate

Pointwise Limits and Uniform Convergence 2

TOPO-XNJWCP

Let $f_{n}(x)$ be a sequence of functions such that $f_{n}(x)=x^{n}$ on the interval $(0,1)$.

Which one of the following statements are TRUE?

A

$f_{n}$ does not converge uniformly to $0$ on $(0,1)$ but does so pointwisely.

B

$f_{n}$ does converge both uniformly and pointwisely to $0$ on $(0,1)$.

C

$f_{n}$ does not converge pointwisely to $0$ on $(0,1)$, but does so uniformly.

D

$f_{n}$ does not converge pointwisely or uniformly to $0$ on $(0,1)$.