Uniform Convergence of Operators implies convergence of their norms

Question

This may be a silly question, but I was asked to prove the following statement:

Let $T_n\to T$ uniformly (here the $T_n$ are bounded linear operators from Banach space $X$ to Banach Space $Y$), by this we mean: $$ \lim_{n\to\infty} \|T_n - T\|_{op} = 0 $$ The question asks me to prove that $\|T_n\| \to \|T\|$. But this seems like a one line proof. Namely, the reverse triangle inequality tells us:

$$ |\|T_n\| - \|T\|| \leq \|T_n - T\| $$ And taking limits (squeeze theorem etc.) on both sides proves the result. Am I missing something?

Answer 1

No, you are not missing anything. That's the standard way of proving that $T_n\to T$ implies $\|T_n\|\to\|T\|$.