So I screwed up a problem on my exam. I know that now. But pure mathematics is as difficult and terrifying as it is rewarding for me, and I can't let this go! If someone could tell me if the following is right or wrong, that would be much appreciated.

I was asked to prove the uniform convergence of the series $f\_n(x)=\frac{x}{x+n}$ on $[1,\infty)$.

So I've taken $$f=\lim\_{n\to \infty} \frac{x}{x+n}=0$$ and $$|f\_n-f|=|\frac{x}{x+n}-0|=\frac{x}{x+n}<1, \forall x\in[1,\infty), \forall n\in\Bbb{N}$$

Then, I simply took $N>0$ and $n \ge N$, concluding that $f\_n(x)$ converged uniformly to $1$.

This seems too easy to be right (so did the solution I put on my exam, God help me!). Is it?

Thanks!