I am not looking for an answer, but rather a hint for the following question: The induced norm $||A||_{a,b}\,$ is defined as $||A||_{a,b} = \max\{||Ax||_b : ||x||_a ≤ 1\}$. Prove that there exists an $x \in \mathbb{R}^n$ such that $||x||_a ≤ 1$ and $||A||_{a,b} = ||Ax||_b$. Hint: Prove that the norm function is continuous and use the extreme value theorem.
I am just really confused on how to begin proving this and was wondering if someone could give me a hint! really just trying to get unstuck.