<s>[INST] Is right the inequality: $|a|+|b| \leq 2|a+ib|$ 
I want know if is right this inequality: $\forall a,b \in \mathbb{R}$,
  $$|a|+|b| \leq 2|a+ib|$$

 [/INST] For real numbers $a$ and $b$ we have
\begin{align*}
&&a^2&\le a^2+b^2\\
&\implies&\sqrt{a^2}&\le\sqrt{a^2+b^2}\\
&&|a|&\le\sqrt{a^2+b^2}
\end{align*}
In the same way we can prove $|b|\le\sqrt{a^2+b^2}$. Since $\sqrt{a^2+b^2}=|a+ib|$, the asked inequality follows.
</s>