Let
$f_n(x)=\frac{x^n}{1+x^n},~x\geq 0, ~n\in \mathbb{N}$
Show that there is no uniform convergence on $[1,+\infty[$.
I found this particular part of an exercise in my textbook and don't know how to approach this.
I can see that it converges to $1$ for $x>1$, but I guess that doesn't help me out here?
Sorry for my lack of work. I'm still unsure about the concept of uniform convergence. The definition says that the "speed" of convergence does not depend on $x$ and I just can't imagine ths concept yet.
Note that for each $n$, $f_n$ is continuous. If $f_n$ converged uniformly on $[1,\infty)$, then this would imply that the limit function $f:=\lim_{n\to\infty} f_n$ is continuous. But $f_n(1)=\frac12$ for each $n$, and $f_n(x)\stackrel{n\to\infty}\longrightarrow 1$ for $x>1$, so $$ f(x)=\begin{cases}\frac12,& x=1\\ 1,&x>1,\end{cases}$$ which clearly is not continuous.