Clueless on how to prove $\|(a,b)\| \le |a|+|b|$

Question

Basically, the inequality says that the norm of a vector is always less than or equal to the sum of the absolute value of its components.

And I know that the norm is defined as:

$$\|(a,b)\|=\sqrt{a^2+b^2}$$

and the absolute value as:

$$|a|=\sqrt{a^2}$$

But I don't know how to relate both equations and prove the inequality.

Answer 1

Squaring gives: $$ ||(a,b)||\leq|a|+|b|\iff (||(a,b)||)^2\leq(|a|+|b|)^2\iff 0\leq 2|a||b|. $$

Answer 2

Simply use the triangle inequality: $\;(a,b)=(a,0)+(0,b)$, so $$\|(a,b)\|\le\|(a,0)\|+\|(0,b)\|=|a|+|b|.$$