Question for finitists -- 0.999... = 1?

[polymath257]

Step 1: Prove the expression .999.... makes sense.

Otherwise, you could argue as follows:

x=1+2+4+8+16+...

2x=2+4+8+16+32+...

Hence,

x=1+2x

so

-x=1

x=-1

In particular,

1+2+4+8+... <0.

Not so, since x = 0.999... has an upper bound as the decimal expansion continues without limit, but x = 1+2+4... diverges.

But it is a perfectly good proof in the 2-adics, where the sum 1+2+4+... does, in fact, converge to -1.