Triangle Inequality on complex numbers

Question

Problem

Let $z= x + iy$, then prove that:

$$|x| + |y| \le 2 ^{1/2} |z|$$

Progress

I've tried to write $|z|$ as $(x^2 + y^2)^{1/2}$, and to make some algebra after this, but I'm really new at proving things, I just get to nothing.

Answer 1

Hints: 1) First square both sides.

2) Write $|z|^2=x^2+y^2$.

3) "Bring the RHS to the LHS": That is- rearrange the inequality in the form $LHS-RHS \geq 0$.

4) Now do an obvious "completing the square" argument and use the fact that the square of any real number is $\geq 0$.

Answer 2

Hint. Since both sides are positive, the inequality is equivalent to $$(|x|+|y|)^2\le2(|x|^2+|y|^2)\ .$$ See if you can simplify this and hence prove that it is always true.

Answer 3

If $z=\Re z+i\Im z$ then not only this one but also you can prove that, $$|\Re z|,|\Im z|≤|z|$$ $$2|\Re z||\Im z|≤|z|^2$$ $$|z|≤|\Re z|+|\Im z|≤√2 |z|$$
All of these are just algebraic manipulations of $(x-y)^2≥0;$ $\forall x,y \in \mathbb{R}.$