[aguaviva]

The simplest argument I can think of is to ask yourself: "Are there any numbers between 0.999... and 1?"

If not then it's logical to conclude they wind up at the same "place", that is, the same number. Or equivalently: it can't have any other value besides 1.

If you care about such things, then you're a mathematician.

[ForOldHack]

It is a limit. A very powerful tool.

You can choose any arbitrary finite number in the sequence, and I can find a number greater. So the value of 1 exists, and its a continuous function, so therefore the limit exists.

I do care about such things, and yes, a degree in math for me was just slogging through every math course the college had, with a long string of high grades, and honor roll achievements.

[floxy]

     1/6 = 0.166666...6
     1/6 = 0.166666...6
     1/6 = 0.166666...6
     1/6 = 0.166666...6
     1/6 = 0.166666...6
  +  1/6 = 0.166666...6
  ----------------------
         = 0.999999...6

...so that 6 times 1/6 is zero point 9 repeating, with a 6 on the end ;-)