By his own actions. He was too focused on revenge to think logically. They have a way to magically show people memories and they knew the Dark Lord was after the Potter's. Sirius should have gone straight to Albus Dumbledore and made a case for why he wasn't the Secret Keeper. The pensieve would have made it clear. And he'd have had the most powerful Wizard of that age to back up his claims. Instead, he chose to take actions into his own hands and respond with violence. If Sirius had succeeded in killing Peter, he would have ended up in the same position, because no one would have believed Sirius's claim in that scenario either.
And, ignoring that part for a moment, if he'd have went to Albus again upon breaking from Azkaban to be like "oh, hey, Peter's a rat;" he'd have likely lived to take Harry on as a ward and saved Harry from a few years of abuse. But instead he was selfish and demonstrated how little he actually cared about the kid.
But...I hate that series anyways so I guess whatever.