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.