Square matrix and induced norm

Let T be any square matrix and let denote any induced norm. Prove that

exists and equals

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

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 must be a lower bound for ?

How do we show that is actually the greatest lower bound? I mean is there an n such that ?