Proof of this relation depends on what approach is your textbook used to learn tensor calculus, for example, If you already defined the action of $g_{\mu \nu}$ for lowering indexes, now you can write $$g_{\mu\nu}g^{\nu\alpha}=g_{\mu\nu}e^\nu \cdot e^\alpha = e_\mu \cdot e^\alpha = \delta^\alpha_\mu$$

Proof of this relation depends on what approach is your textbook used to learn tensor calculus, for example, If you already defined the action of $g_{\mu \nu}$ for lowering indexes, now you can write $$g_{\mu\nu}g^{\nu\alpha}=g_{\mu\nu}e^\nu \cdot e^\alpha = e_\mu \cdot e^\alpha = \delta^\alpha_\mu$$

This proof is in Core Principles of Special and General Relativity by Luscombe: