Why is 0.99999 equal to 1?

This is an old standby, recurring (if you'll excuse the pun) on a regular basis in many, many forums. It never seems to be settled to everyone's satisfaction, so I'd bet there are people here on HN who don't believe that 0.999999... is equal to 1.

I would be interested in engaging with a few, in part to help them see why modern mainstream mathematics says that, in part to try to help me understand their point of view, and in part to test an idea I have about a discussion system.

I'm intending this to take part over a few weeks, and I won't always be able to respond instantly to questions, so it would favour anyone interested in a thoughtful discussion. If you're interested, please feel free to send me an email - details are in my profile.

Thanks.