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.