Checking for Uniform Convergence

Question

Check the Uniform Convergence of $\large f_n(x)=x-{x^n\over n}$ in $[0,1]$

I have problem in the very first step in showing its point wise convergence.

At $x=0, f(x)=0$ but what happens at other places ? it wont be $0$ right ?

Answer 1

The sequence $(f_n)$ is pointwise convergent to the function $f$ defined by $f(x)=x$ on the interval $[-1,1]$ and since $$\forall x\in[-1,1],\quad |f_n(x)-f(x)|=\frac{|x|^n}{n}\leq \frac{1}{n}\to 0$$ the the sequence $(f_n)$ is uniformly convergent to $f$ on $[-1,1]$.