Operator Norm on Banach Space is Submultiplicative

Theorem
Let $B$ be a Banach space, and let $S, T \in \mathfrak{L}(B, B)$ be bounded linear operators on $B$. Then $\|ST\|_* \leq \|S\|_* \|T\|_*$, where $\|\cdot\|_*$ is the operator norm.

Proof
Let $x\in B$ have norm $1$. Then

Taking the supremum of all $x\in B$ with norm $1$, the left-hand side of the above inequality becomes $\|ST\|_*$.