Solution. Note that

$$

0 \leq(|\operatorname{Re} z|+|\operatorname{Im} z|)^{2}=|\operatorname{Re} z|^{2}-2|\operatorname{Re} z||\operatorname{Im} z|+|\operatorname{Im} z|^{2}

$$

Thus

$$

2|\operatorname{Re} z||\operatorname{Im} z| \leq|\operatorname{Re} z|^{2}+|\operatorname{Im} z|^{2}

$$

and then

$$

|\operatorname{Re} z|^{2}+2|\operatorname{Re} z||\operatorname{Im} z|+|\operatorname{Im} z|^{2} \leq 2\left(|\operatorname{Re} z|^{2}+|\operatorname{Im} z|^{2}\right)

$$

That is

$$

(|\operatorname{Re} z|+|\operatorname{Im} z|)^{2} \leq 2\left(|\operatorname{Re} z|^{2}+|\operatorname{Im} z|^{2}\right)=2|z|^{2}

$$

and therefore,

$$

|\operatorname{Re} z|+|\operatorname{Im} z| \leq \sqrt{2}|z|

$$