the number of bitcoins in existence will never exceed 21 million.
You are correct.
As a matter of fact, the number of bitcoins will actually never exceed 20999999.97690000 BTC.
This will happen somewhere around the year 2140 (about 126 years from now).
Of course during this time, the block subsidy will be reduced by half (and rounded down to the nearest 0.01 microbitcoins) every 4 years. So the 25 BTC subsidy that miners receive today will be reduced to 12.5 BTC in about 3 years. Then that will be reduced to 6.25 BTC 4 years later, and so on.
In the meantime, the value of the total transaction fees of all transactions included in the block will increase over time as bitcoin gains popularity. Eventually the subsidy will shrink to an amount that is less than the transaction fees. At that time, the reward the miners receive will mostly come from transaction fees instead of the subsidy.
When the subsidy eventually shrinks to 0 BTC, mining will be entirely supported by transaction fees.
