How to show proof of income for tax purposes?

You can't prove something that doesn't exist dear.

Where did you get money to live on?

How did you pay rent, food, utilities, etc.?

How do you show proof of something that doesn't exist?   You don't.

As far as your tax return goes, just fill it out correctly and send it in.   No "proof" of anything is required unless the IRS comes and asks you for it.

You file a 1040 tax return every year, and that documents your income / no income. {Logically, you cannot prove income if there is none.}

You don't.  Ever.

When you do your taxes, you MUST keep the proof in your own possession.  You do NOT show it.  You simply report what you claim your income was, but there is no opportunity to show any proof.  In the rare event that proof is required, it will be requested much, often not for several years.  The worst thing you can do is to send the proof when you do your taxes, because then you won't have it when they do ask for it.

Literally just write on a piece of paper your income and where it was from.