Let $(Y,h)$ be a compactification of a compact Hausdorff space $X$, prove that $h(X)=Y$.
My attempt. Since $(Y,h)$ is a compactification of $X$, we have that $Y$ is a compact space and the function $g: X \to h(X)$ will be a homeomorphism and $h(X)$ is dense in $Y$, then $$\overline{h(X)}=Y.$$ I have to prove that $h(X)=Y$, my idea is to prove that $h(X)$ is closed in $Y$ and we have that $h(X)= \overline{h(X)}$. Since $X$ is compact and Hausdorff we have that $h(X) is also compact and Hausdorff. I have tried to prove that h(X) is closed but I am stuck here. Any suggestions or should I take another way?
Continuous image of a compact space is compact and compact subsets of Hausdorff spaces are closed.