How to prove the equivalency of the following two optimization problems?

Question

The following two problems are equivalent

(P1) $\underset{\mathbf x}{\arg\min}\,\, f(\mathbf x)+g(\mathbf x)$

(P2) $\underset{\mathbf x,y}{\arg\min}\,\, f(\mathbf x)+y\quad \quad \text{s.t.}\,\,\, g(\mathbf x)\leq y$

I want to know how to prove it rigorously.

Thanks for any help.