Someone’s lifestyle proves a fact. Personal confirmation. Believing what one’s eyes show. ‘Proof’ implies a known fact ,so living proof refers to a person’s lifestyle or exploits that prove a good or bad factoid. For example ‘”Obama is living proof a non white can become President” but just as true is “The resurgence of white supremacists in America is living proof of the Obama presidency aftermath”.
“My uncle is living proof that you can be a successful and alcoholic.”