# 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 .

Title uniqueness of limit of sequence UniquenessOfLimitOfSequence 2013-03-22 19:00:23 2013-03-22 19:00:23 pahio (2872) pahio (2872) 4 pahio (2872) Theorem msc 40A05