limit of real number sequence

An endless real number sequenceMathworldPlanetmath

a1,a2,a3, (1)

has the real number A as its limit, if the distancePlanetmathPlanetmath between A and an can be made smaller than an arbitrarily small positive number ε by chosing the n of an sufficiently great, i.e. greater than a number N (the of which depends on the value of ε); accordingly


Then we may denote

limnan=A (2)

or equivalently

anAasn. (3)

Remark 1.  One should not think, that  an=A  when  n=.  The symbol “” no number, one cannot set it for the value of n.  It’s only a question of allowing n to exceed any necessary value.

Example 1.  Using the notation (2) we can write a result

limn2nn+1= 2.

It’s a question of that the real number sequence


has the limit value 2 (e.g. the nine hundred ninety-ninth member  19981000=1.998  is already “almost” 2!).  For justificating the result, let ε be an arbitrary positive number, as small as you want.  Then

|2-2nn+1|=|2n+2n+1-2nn+1|=|2n+1|=2n+1<ε, (4)

when n is chosen so big that

n>2ε-1. (5)

The condition (5) is obtained from (4) by solving this inequality for n.  In this case, we have  N=2ε-1.

Example 2.  The so-called decimal expansions, i.e. endless decimal numbers, such as

3.14159265=π,0.636363,0.99999, (6)

are, as a matter of fact, limits of certain real number sequences.  E.g. the last of these is related to the sequence

0.9, 0.99, 0.999, (7)

which may be also written as

1-110, 1-1102, 1-1103,

The limit of (7) is 1.  Actually, if  ε>0,  the distance between 1 and the nth member of (7) is


when  10n>1ε,  i.e. when  n>-log10ε=N.

The endless decimal notations (6) and others are, in fact, limit notations — no finite amount of decimals in them suffices to give their exact values.

Remark 2.  In both of the above examples, no of the sequence members was equal to the limit, but it does not need always to be so; thus for example


and every other member of the sequence in question is 0.

Infinite limits of real number sequences

There are sequences that have no limit at all, for example  1,-1, 1,-1, 1,-1,.  Some real number sequences (1) have the property, that the member an may exeed every beforehand given real number M if one takes n greater than some value N (which depends on M):


Then we write


Similarly, the sequence (1) may be such that for each positive M there is N such that


and then we write



Title limit of real number sequence
Canonical name LimitOfRealNumberSequence
Date of creation 2015-01-30 17:53:27
Last modified on 2015-01-30 17:53:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Definition
Classification msc 40A05
Synonym limit of sequence of real numbers
Related topic GeometricSequence
Related topic BriggsianLogarithms
Related topic InfiniteProductOfSums1a_i
Defines limit