May someone please tell me if the following proof is correct? Also, may someone please tell me what I could possibly do to improve the structure and proof in general?
What I want to prove: Let Let X and Y be metric spaces. Assume X is a discrete metric space. Then $f: E_{discrete} \rightarrow Y$ is continuous.
My proof: Let p$\in E_{discrete}$. Let $\epsilon >0$. What I want to show is that $\forall \epsilon >0$ $\exists \delta >0$ : $\forall x\in E$ if $x \in N_{\delta}(p)$ then $f(x) \in N_{\epsilon}(f(p))$. Set $\delta = \frac{1}{2}$ then for $x\in E_{discrete}$ if $d(x,p)<\delta$ then clearly $d(x,p)=0$ and so $d(f(x),f(p))=0 < \epsilon$. Is my proof correct? What can I do to improve it? I've heard that the result follows vacuously but i'm not sure why that is the case. Isn't a vacuous truth a truth for when a statement p is false and q in true in a statement p $\implies q$ ?