Submultiplicity of Induced Norms

Question

This is a lecture note on induced norms from Cornell: pdf

Given $A$ is a matrix and $v$ is a vector with length less than or equal to $1$. It says, "if $||\cdot||$ is an induced norm, then $||Av||\leq ||A||\cdot||v||$ from the definition of vector norms."

I check all definitions on Wikipedia, but could not figure out why. Is there any hint or comment? I don't think vector norms have such properties.

Answer 1

Hint:

Let me write your inequality when $v \ne 0$ as

$$\frac{\|Av\|}{\|v\|}\le \|A\|$$

Answer 2

Nah, I think they just mean 'from the definition of induced norm'. Since the norm of $A$ is defined as the supremum, we have inequality for all $v \in V$.