Are you required to pay first month, last month and security deposit? As a landlord I would find the security deposit far more important than proof of income.
I think you are dealing with a paranoid landlord. I don't think proof of income is a reliable indicator regarding what kind of tenant you will get. I know a lot of people who earn a lot, but don't save much and vice versa.
As long as you hold the first, last and security deposit...it is in everyone's interest to make the agreement work out and the earnings of the tenant become mostly irrelevant.
Most of my previous landlords have asked for income proof, I don't think it's a big deal at all. Makes sense from a landlord's point of view - they want to have some idea of their tenant's financial standing. Yes you can argue that income alone doesn't provide a complete financial picture, but it's better than nothing and certainly less invasive than requesting to see a bank balance.
Income alone is very poor predictor of credit worthiness. Debt to income ratio is far more important. So unless your landlord can verify what debt you are holding, it matters very little what your income is.