Anybody can buy and build? Yeah... only if they have the funds for it!
That's what you're ignoring: running out of funds isn't just a possibility for the monopoly landlord, it's possible for new entrants to the market, as well as the construction companies. The new entrants and construction companies are facing that risk, just as the monopoly landlord is.
For example, a new entrant to the market may have to go into debt to fund his bid or construction in order to compete with the landlord. The monopoly landlord on the other hand, might not have debt and instead has large capital reserves. Depending on the circumstances, it is even possible for the new entrant needing to rent ABOVE what the monopoly landlord already does to just to break even on costs that the monopoly landlord doesn't have (e.g. because of debt, and/or they overbid for the risk in the market, higher land costs, and other reasons). If done in a market where there is already more supply than renters, it would be a very risky venture for the new entrant.
And again, it's also entirely possible that there are 0 bidders showing up to compete with the monopoly landlord (due to shear lack of other participants willing to step in to compete). In which case, the monopoly landlord doesn't have just a monopoly on rents, but bids, and the construction companies would have tough time running the monopoly landlord out of business, no matter how much they build.
> It costs them $200,000 to build something the monopoly landlord has to buy from them for a million dollars, or someone else will buy it from them for $999,999 which then happens again and again and destroys the monopoly
Look... we can make a bunch of "what if" scenarios, but the scenario you made doesn't necessarily have to be the case (nor any of my examples). But, you had inquired about:
>> You cannot realistically build enough so that the rents decrease.
>I'm curious what makes you think this.