Prove that Matrix Norm (1 norm) and Vector Norm (1 norm) are Compatible

Question

I need to show that, the matrix norm ||A||1 and the vector norm ||v||1 are compatible, that is,

||Av||1 ≤ ||A||1 ||v||1 ,∀v∈Cn ∀A∈Cnxn

I have encountered this, but I don't understand what the writing underlined in red means:

Might help:

Answer 1

For a induced matrix norm it holds by definition: $$||A|| := \sup_{x\not= 0} \frac{||Ax||}{||x||}$$

Hence it follows for all $x\not= 0$:

$$\frac{||Ax||}{||x||} \le \sup_{x\not= 0} \frac{||Ax||}{||x||} =: ||A||$$

Multiplying both sides with $||x||$ gives the wanted result for $x\not=0$: $$||Ax|| \le ||A||\cdot||x||$$

The relation also holds obviously for $x=0$ hence for all $x$.