Given an operator $T\in B(X,X)$, I'm trying to prove that

$$\lim\_{n\to\infty} \left( ||T^n||\right)^\frac1n \leq ||T|| $$

I can show that $||T^n||\leq||T||^n$, but only for finite $n$. How can I be sure that this holds in the limit? I'm a bit confused as to what kind of continuity that would require.