[ColinWright]

That doesn't really work, because you have to be able to hit them all simultaneously, and in fact, you can't. The theorem says that for any given epsilon you can choose enough to ensure that you are within epsilon everywhere except in a very small section of the real line, and you can make the parts where it's not within epsilon very small.

The problem with your comment is that yes, for any given point you can get as close as you like, but the hard part is getting close nearly everywhere all at once. More, you then have to show that there's some sort of convergence, and that where you are close now is a superset of where you are close when you demand to be closer.

It's not straight forward.