Sequences and Limits of Functions

Thedefinition of continuity:

A functionis continuous at a pointif forgiven there existssuch thatIn this definitionis generally a function of

may be restated in terms of sequences.

We need first a definition.

Definition Letbe a set of real numbers. A real numberis an accumulation point ofif and only if every neighbourhood ofcontains infinitely many points of

Theorem

Letwithan accumulation point ofThenhas a limit atif and only if for each sequenceconverging towithinandfor allthe sequenceconverges.

Proof: Suppose thathas a limitat x-0 . Letbe a sequence of members ofdistinct frombut converging toand consider the sequenceChoose There issuch that ifwiththen Sinceconverges tothere is N such that for n>=N abs {x-n -x-0} < %delta . For n>=N, 0<abs {x-n -x-0} <%delta and x-n in D, henceThenconverges to L.

Suppose now that the latter condition is satisfied. All the sequenceshave a common limit, because if there are two sequences that do not converge to the same limit, sayconverges toandconverges tothen we can define a new sequencesuch thatandThis sequence consists of members ofdistinct fromand converges tohenceconverges hencesinceandare subsequences of this convergent sequence.

Suppose all the subsequences ofconverge to a limitSuppose thatis not a limit ofatthen there issuch that for everythere iswithandIn particular for eachthere iswith such thatThe sequenceconverges toand is a sequence of members of distinct fromhencetocontrary to hencemust be the limit ofat