For any complex number `z` prove that `|R e(z)|+|I m(z)|<=sqrt(2)|z|`

Explanation

To prove that for any complex number \( z \), the inequality \( | \text{Re}(z) | + | \text{Im}(z) | \leq \sqrt{2} |z| \) holds, we can follow these steps:

### Step-by-Step Solution:

1. **Express the Complex Number**:
Let \( z \) be represented as \( z = x + iy \), where \( x = \text{Re}(z) \) (the real part) and \( y = \text{Im}(z) \) (the imaginary part).

2. **Calculate the Modulus of \( z \)**:
...