I am trying to prove that $\{f_{n}(x)\}_{n\in \mathbb{N}}$, where $f_{n}(x)=(x+\frac{1}{n})^2$, and $x\in \mathbb{R}$, does not converge uniformly to $f(x)=x^2$, $x\in \mathbb{R}$. Is the following correct?
Set $\epsilon=2$. Consider arbitrary $N\in \mathbb{N}$. Consider arbitrary $n\in \mathbb{N}$ such that $n>N$. Set $x=n$. Then, we have that: $$|(x+\frac{1}{n})^2-x^2|= |(n+\frac{1}{n})^2-n^2|=|n^2+2+\frac{1}{n^2}-n^2|=|2+\frac{1}{n}|>\epsilon$$ where the second equality follows from expanding the square on the LHS.