$T$ is bounded iff $S\circ T$ is bounded. Let $X , Y, Z$ are Banach spaces. $T:X\to Y$ be a linear map and $S:Y\to Z$ be a one-one linear map which is bounded. Show that $T$ is bounded iff $S\circ T$ is bounded.

One side is trivial that I know. But how to prove that if $S\circ T$ is bounded then $T$ is bounded too.
I think I got this one.

Since $X, Y, Z$ are Banach spaces, so we can use the Closed Graph theorem. We want to show $T$ is bounded. We prove $Gr(T)$ is closed. So take $(x_{n}, T(x_{n})) \to (x,y)$. That means $x_{n} \to x$ and $T(x_{n})\to y$. We need to prove $T(x)=y$. 

Now we have since $S\circ T$ is bounded $S\circ T(x_{n}) \to S\circ T(x)$. Again since $S$ is bounded we have $S(T(x_{n})\to S(y)$. So by uniqueness of limit we have $S(T(x))=S(y)$. Since $S$ is one-one $T(x)=y$.

Hope this is fine!
The result is true if $S$ is bijective, otherwise it is false, $S$ is invertible and bounded implies that $S$ is open by the open mapping theorem. We deduce that $S^{-1}$ is continuous, hence bounded. Write $U=S\circ T$, and suppose that $U$ is bounded, we have $T=S^{-1}\circ U$ is bounded as product of two bounded maps.