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


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

|an-a|<|a-b|2for alln> some n1


|an-b|<|a-b|2for alln> some n2

But when n exceeds the greater of n1 and n2, we can write


This inequalityMathworldPlanetmath 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