Question

Question: is 1 > 0.99999999...?
Prove algebraically.

Answer 1

No. We claim that
`1=0.\overline{9}` and use algebra to prove the statement.

Let
`x=0.\overline{9}`. Multiply this by ten to get
`10x=9.\overline{9}`. Subtract the initial equation to give
`9x=9` and divide by
`9` to see that
`x=1`. Substituting into the original equation gives
`1=0.\overline{9}`, proving the desired statement.