\(x\quad =\quad 0.999999...\\ 10x\quad =\quad 9.99999...\\ 10x\quad -\quad x\quad =\quad 9.000....\\ x=1\)

Or

\(0.9999...\quad =\quad \frac { 9 }{ 10 } *{ \frac { 1 }{ 10 } }^{ 0 }+\frac { 9 }{ 10 } *{ \frac { 1 }{ 10 } }^{ 1 }+....\frac { 9 }{ 10 } *{ \frac { 1 }{ 10 } }^{ n }\\ \\ Using\quad infinite\quad sum\quad formula\quad for\quad a\quad geometric\quad series:\\ \\ \frac { \frac { 9 }{ 10 } }{ 1-\frac { 1 }{ 10 } } \quad =\quad \frac { \frac { 9 }{ 10 } }{ \frac { 9 }{ 10 } } \quad =\quad 1\)

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