Let $X$ and $Y$ be two metric spaces and $B(X,Y)$ the metric space of the bounded functions from $X$ to $Y$. Then, the closure of an equicontinuous subset of $B(X,Y)$ is equicontinuous.
I have the following solution (simplified to leave the thing I have doubts about):
Let $A$ be an equicontinuous subset of $B(X,Y)$, then if $f$ in the closure of $A$, exists $g$ in $A$ such that $d(f,g) \lt \epsilon/3$. Since $A$ is equicontinuous, exists $\delta \gt 0$ such that $d(x,y) \lt \delta \Rightarrow d(g(x),g(y)) \lt \epsilon/3$, so if $d(x,y) \lt \delta$:
$$d(f(x),f(y)) \leq d(f(x), g(x)) + d(g(x),g(y)) + d(g(y),f(y)) \lt \epsilon$$
The doubt I have is, since I'm using that $A$ is equicontinuous taking a $\delta$ such that the distance between two points in $Y$ is less than $\epsilon/3$ shouldn't I take a $\delta$ that make the same thing (being less than $\epsilon/3$).
I know that normally this doesn't happen, but here I think it should be wrong doing this, since I'm taking the same delta for different $\epsilon$ proving different things.