Let $Y$ be a closed subspace of normed space $X$. Prove that there exists $f \in X^*$ such that $f(Y)=0$ and $\|f\|=1$

Question

Let $Y$ be a closed subspace of normed space $X$. Prove that there exists $f \in X^*$ such that $f(Y)=0$ and $\|f\|=1$

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I think Hahn-Banach Theorem must be used. But $f(Y)=0$ and $\|f\|=1$ come from different results of Hahn-Banach Theorem . I have tried to prove it by contradiction. It also doesn't seem working. Could someone help me please?