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
Canonical name	UniquenessOfLimitOfSequence
Date of creation	2013-03-22 19:00:23
Last modified on	2013-03-22 19:00:23
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	4
Author	pahio (2872)
Entry type	Theorem
Classification	msc 40A05