Show uniform convergence of the sequence $f_n(x) = ( x-x^n/n)$ for $x \in [0,1]$ Show uniform convergence of the sequence $(x-x^n/n)$ on $[0,1]$
To start with, i am not even able to see what the point was limit of this sequence should be. Little confused. like for $x=0$ I think the limit should be $0$ and for $x=1$ it should be $1$ and for values in between i think it should be $x$. Is that correct ? How do i further show its uniform convergence ?

The sequence of functions $f_n$:  $\left(x-\frac{x^n}n\right)_n$ is point-wise convergent to the function $f\colon x\mapsto x$ and we have
$$\forall x\in[0,1],\quad|f_n(x)-f(x)|=\frac{x^n}n\le \frac1n\xrightarrow{n\to\infty}0$$
so the convergence is uniform.
