Proving Uniform Convergence of a Sequence3

Question

Show that this sequence $\large f_n(x)={x^n\over n}$ on

a) $\left]-\infty,\infty\right[ $ is not uniformly convergent.

I have tried showing that the given sequence doesn't converge in this interval and hence not uniformly convergent.

b) $[0,1]$ is uniformly convergent.

I have reached this far :

$$|f_n(x)-f(x)|=\left|{x^n\over n}-0\right|={x^n\over n}<\epsilon$$ What should i do next ?

Answer 1

a) Yes since $(f_n)$ isn't convergent on $\mathbb R$ so itsn't uniformly convergent.

b) To complete:

$$\forall x\in[0,1],\quad|f_n(x)-f(x)|=|{x^n\over n}-0|\leq \frac{1}{n}\to0$$

Answer 2

Hint: for b)

what is $\lim n \to \infty $ of $x^n$ on $[0,1]$ ? Can you find the pointwise covergence on your own after knowing this limit?

unfiorm convergence : what is the $\sup$ of $ \ \dfrac{x^n} {n } $ on $[0,1]$ ?