Square matrix and induced norm

Let T be any square matrix and let $\displaystyle \left\| \cdot \right\|$ denote any induced norm. Prove that

$\displaystyle lim_{n \rightarrow _{\infty}} \left\| T^{n} \right\| ^{1/n} $ exists and equals $\displaystyle inf _{n=1,2,\cdots } \left\| T^{n} \right\| ^{1/n} $

I am not sure how I go about proving that the limit exists.

For the infimum, I think it has something to do with the fact that

$\displaystyle \left\| T^{n} \right\| = sup_{x \neq 0} \frac{\left\| T^{n} x\right\|}{\left\| x \right\|} $

But I don't know how this information helps with the solution of the problem.

Square matrix and induced norm

I might be wrong but according to that proof, does this just mean that $\displaystyle \rho (T)$ must be a lower bound for $\displaystyle \{ \left\| T^{n} \right\| ^{1/n} : n,1,2, \cdots \}$ ?

How do we show that $\displaystyle \rho (T)$ is actually the greatest lower bound? I mean is there an n such that $\displaystyle \rho (T) = \left\| T^{n} \right\| ^{1/n} $ ?