Lambert W of Zero is Zero

Theorem
Let $W_0: \left[{-\dfrac 1 e, \infty}\right) \to \left[{-1, \infty}\right)$ denote principal branch of the Lambert W function.

Then:
 * $W_0 \left({0}\right) = 0$

Proof
From the definition of the principal branch of the Lambert W function:
 * $y = W_0 \left({x}\right) \iff x = y e^y$

where $x \in \left[{-\dfrac 1 e, \infty}\right)$ and $y \in \left[{-1, \infty}\right)$.

The result follows from substituting $x = 0$ and $y = 0$.