Show that $T$ is bounded if $fT$ is linear functional

Let $X$ and $Y$ be Banach space. If $T: X\to Y$ is a linear map so that $f(T)\in X^*$ (dual space of $X$) for every $f\in Y^*$, then $T$ is bounded.

Here is the proof by contradiction of this question, but I am confused where is the contradiction.

Let $x_n\to x$ in $X$ and $Tx_n\to y$ in $Y$. It is enough to show that $Tx=y$ by closed mapping theorem.

If not, $Tx\neq y$. Then by a corollary of the Hahn-Banach theorem,

Let $Y$ be a closed subspace of a normed space $X$, $u\in X\setminus Y$ and $\rho=dist(u,Y)$. Then there exists a linear functional $f\in X^*$ so that $f(x)=1$ and $f|Y=0$, and $\|f\|=\rho^{-1}$.

we take $Y=\{0\}$ and then there exists $f\in Y^*$ so that $f(Tx-y)=1$, where is a contradiction.

I am confused why that is a contradiction.