That begs the question, what happens to methane to limit its greenhouse lifetime?
It's not pretty. Essentially, the C-H bonds in methane are vulnerable to radical reactions. This allows for a variety of removal processes, many leading to the formation of water vapour and/or CO2 itself.
While that may not sound so bad, don't forget that water vapour is one of the most powerful greenhouse gases when it's found in the atmosphere, which is why, for example, the effective carbon emissions of intercontinental flights are so significant. So the end result is methane, an awful greenhouse gas, lives a relatively short life but ends up as either a worse or slightly less awful different greenhouse gas. In other words, methane stinks!