$f_n(x) = x - x^n$ for $x\in [0,1]$. Does the sequence converge pointwise or uniformly on $[0,1]$?
We also have to prove why it is or isn't pointwise or uniformly convergent.
I have worked out that on $x\in[0,1)$, the limit as $n$ approaches infinity is $x$. And when $x=1$ the limit is $0$.
So this means its pointwise convergent right? But how do I prove this? Also how on earth do I prove it is not uniformly convergent? (I'm just assuming it's not).
If $0\le x<1$, you have $\lim_{n\to\infty}x^n=0$. Moreover $f_n(1)=0$. So you have, by basic rules of limits, $$ \lim_{n\to\infty}f_n(x)= \begin{cases} x & \text{if $x\in[0,1)$}\\[4px] 0 & \text{if $x=1$} \end{cases} $$ which is pretty much what you did. So the sequence is pointwise convergent.
It is not uniformly convergent, because a uniformly convergent sequence converges also pointwise to a continuous function.