Showing uniform convergence of sequence $\langle\frac{x^n}n\rangle$

Question

Show that the sequence $f_n(x)=\frac{x^n}n$ on $[0,1]$ converges uniformly to $0$.

I am able to show that the given sequence converges point-wise to $0$. Now, trying to using the Cauchy's principle to show uniform convergence i am unable to find a suitable $m$. Need some hint please.

Answer 1

What we need is $\sup_{x\in[0,1]}|f_n(x)-f(x)|\to 0,n\to\infty$

Now $\sup_{x\in[0,1]}|f_n(x)-f(x)|=\sup_{x\in[0,1]}|\frac{x^n}{n}|\le\frac{1}{n}\to 0,n\to\infty$

Thus we have uniform convergence.