The sequence of functions $\{x,x/2,x/3,...\}$ is not uniformly convergent

Question

I want to show the sequence of real functions $(f_n)$ where $f_i(x)=\frac{x}{i}$ is not uniformly convergent, though it converges pointwise to $f=0$.

Here's my solution:

Let $\varepsilon=1$. Then given any $N$, we can let $n=N+1$ and $x=N+2$. Then $|\frac{x}{n}|=\left|\frac{N+2}{N+1}\right|>1$. Thus $|f_n-0|>\varepsilon$ and $(f_n)$ is not uniformly convergent.

My question is whether I can let $x$ depend on $n$ like that, and whether what I did was valid.

Thanks!

Answer 1

Your solution is fine. Here's, the same idea, with (in my opinion) an easier presentation.

Suppose the sequence converges uniformly. Then, for all $n$ sufficiently large, $$\frac{1}{n}\sup_{x\in\mathbb{R}}\left|x\right|=\sup_{x\in\mathbb{R}}\left|\frac{x}{n}-0\right|<1$$ This is clearly a contradiction, since $|x|$ is unbounded.

This should answer your question as well: "$x$ can depend on $n$" (but not the other way around).