Prove that $f_{n}(z)=\frac{z + n}{n}$ does not converge uniformly

Question

Let $(f_n )_n$ be a sequence of functions $f_n:U\rightarrow \mathbb{C}$, with $f_{n}(z)=\frac{z + n}{n}$ and $U \subset \mathbb{C}$.

I think I understand the concept of uniform convergence but I'm struggling to prove that this sequence does not converge uniformly. How can I do that?

Answer 1

Your sequence of functions converges pointwise to $1$. If $U\supset\Bbb N$, then $(\forall n\in\Bbb N):f_n(n)=2$, and therefore $(f_n)_{n\in\Bbb N}$ does not converge uniformly to $1$. Of course, for certain sets $U$ it does.