Question

Show that {fn}, where fn : [0, 1] → R is the map fn(x) = x
n, is not uniformly convergent.

Answer 1

Given $f_n: \ [0,1] \to R$ such that $f_n(x) =x^n$ .

Now, Pointwise convergence is $\lim_{n\to \infin} f_n(x) = \lim_{n\to \infin} x^n = \begin{cases} 0 \ if \ x\in[0,1) \\ 1 \ if \ x=1 \end{cases}$ .

Each $f_n(x) =x^n$ is continuous for every n but it's pointwise limit is not continuous. So $f_n(x)$ is not uniformly convergent.