uniqueness of limit of sequence

If a number sequence has a limit, then the limit is uniquely determined.

Proof. For an indirect proof (http://planetmath.org/ReductioAdAbsurdum), suppose that a sequence

a_{1},\\,a_{2},\\,a_{3},\\,\\ldots

has two distinct limits $a$ and $b$ . Thus we must have both

|a_{n}\\!-\\!a|<\\frac{|a\\!-\\!b|}{2}\\quad\\mbox{for all}\\;\\;n>\\mbox{\\,some\\;}n_{1}

and

|a_{n}\\!-\\!b|<\\frac{|a\\!-\\!b|}{2}\\quad\\mbox{for all}\\;\\;n>\\mbox{\\,some\\;}n_{2}

But when $n$ exceeds the greater of $n_{1}$ and $n_{2}$ , we can write

|a\\!-\\!b|\\;=\\;|a\\!-\\!a_{n}\\!+\\!a_{n}\\!-\\!b|\\;\\leqq\\;|a\\!-\\!a_{n}|+|a_{n}\\!-\\!b%|\\;<\\;\\frac{|a\\!-\\!b|}{2}+\\frac{|a\\!-\\!b|}{2}\\;=\\;|a\\!-\\!b|.

This inequality an impossibility, whence the antithesis made in the begin is wrong and the assertion is .