A version of the Closed Graph Theorem states that if $T: X\rightarrow Y$ is a linear operator between Banach spaces then $T$ is bounded iff the graph of $T$ is closed in $X\times Y$.
To check if the graph is closed we suppose that a sequence $x_n \rightarrow x\in X$ and that $Tx_n \rightarrow y\in Y$ and then have to show that $x\in X$ and $Tx = y$.
My question is, what if $x_n\rightarrow x\in X$, but $Tx_n$ doesn't converge. Then the graph of $T$ is not necessarily closed, but $T$ is necessarily unbounded. Doesn't this contradict the theorem?
I'm sure I'm misunderstanding something simple - any help would be greatly appreciated!