Closed graph theorem( metric spaces)

Question

So what I want to prove is the following:

If $f:X \to Y$ is a continuous function and $X$ and $Y$ are metric spaces, then the graph of $f$ is closed.

All the proofs that I’ve encountered in the internet involve some topological concepts that I’m not aware of. I was looking for a proof which would involve only concepts from metric spaces. Any suggestions would be really helpful.

Answer 1

Take a look at a sequence $(x_n,y_n) \in \text{graph}(f)$ converging in $X\times Y$ to some $(x,y)$. Since we look at the graph of f, we have $y_n=f(x_n)$. We have to prove, that $(x_n,y_n)$ converges in $\text{graph}(f)$. Since $x_n\rightarrow x$ and f is continuous, $y_n=f(x_n)\rightarrow f(x)=y$, thus the limit of our sequence is $(x,y)=(x,f(x))$ and our sequence converges in $\text{graph}(f)$. Thus $\text{graph}(f)$ is closed.