Proof Verification: $f(\Lambda_n(x))$ is pointwise bounded for all $x$, and all $f \in X^* $then $\Lambda_n$ is uniformly bounded

Question

I found the question here: If $\Lambda_n \in B(X,X)$ and $f(\Lambda_n(x))$ is pointwise bounded for all $x$, and all $f \in X^* $then $\Lambda_n$ is uniformly bounded

I think I came up with a proof, but i am not certain it is correct. By UBP we know that $f(\Lambda_n)$ is uniformly bounded for each $f \in X^*$. Now by banach theorem we know that for each $x$ there is $f\in X^*$ s.t $f(x)=\|x\|$. Thus for each $x$, $\|\Lambda_n(x)\|$ is bounded and so by UBP again we get $\Lambda_n$ are uniformly bounded.