If Banach space is dense in anothe Bamach space, will the dual space dense as well?

Question

Let $X$ and $Y$ be Banach spaces. $X\subset Y$ is dense in $Y$. Prove or disprove: $Y^*$ is dense in $X^*$.

I tried as follows. Since $X\subset Y$, then $Y^*\subset X^*$. Let $f\in X^*\setminus Y^*$. I tried to construct a sequence $f_n\in Y^*$, $f_n\to f$, but failed.