How do you prove unemployment history?

Are you collecting unemployment benefits? If so, they will have your records.

Good question, because it is very hard to prove a negative.

Do you still have your unemployment checks/vouchers/documents (depends on what state you live in) ? Those would establish the 20 weeks for sure. Although they could probably get that from the unemployment state offices.

I’m not sure how you could prove that you are not employed now.

No, I am not currently or have ever received unemployment benefits

Have you filled out any applications during this time where you marked that you were unemployed? Maybe those could be used.
[Addition]: I know this is more personal, but what about bank/tax documents? If there’s no significant income, then clearly you have not been employed, right?

How about Social Security taxes? If you receive wages or salary, you typically pay Social Security taxes (which are withheld from your paycheck). So, if you haven’t paid these taxes, then presumably you’ve been unemployed? Contact the Social Security Administration.

But this doesn’t address the issue of self-employment. If you’re self-employed, or have been, are you still eligible for the training course?

What were you doing when you were not employed? Going to school, being a stay-at-home house person or parent? Perhaps you could document what you were doing?

You could get a transcript of your income taxes from the IRS to show what you have had for income the last few years.

1) have been unemployed for at least 20 weeks in the past 5 years

Give them enough of a schedule of your employment in the last 5 years that shows gaps that add up to at least 20 weeks. Might want to label the gaps too. Provide who you were employed by, and their contact info. They probably don’t need more evidence but if you want, you could also show a relevant page from your IRS records from some year(s) that show such gaps, listing the employers you list but no others.

2) be currently unemployed.

List your previous employer, last date employed, and their contact information.

@Adirondackwannabe Has it right. The IRS can give you a printout of your recorded income that will show a gap during the period you refer to. It’s for free, but it will require a visit to their nearest office, for two reasons: They are nearly impossible to contact by phone, and they need to see you, your face, and a corresponding picture ID, and Social Security card before they will hand it over to you.