$A$ closed in $Y$ and $Y$ closed in $X$, then $A$ is closed in $X$

Question

This answers proves it, but what's wrong with saying that a closed set $A$ of $Y$ is the form:

$$A = Y\cap C$$

where $C$ is a closed set in $X$. Then, since both $Y$ and $C$ are closed in $X$, so is their intersection. Then $A$ is closed in $X$.

?

Answer 1

That's a fine way to say it too. The two arguments are essentially the same; any statement about closed sets can be equally well phrased in terms of open sets by taking the complements of everything, and that's what the linked answer does to your argument.