The Equivalence of the Epsilon - Delta and the Sequenctial Definitions of Continuity

Thedefinition of continuity states: If for a functionwiththen givensuch thatthere existssuch that ifthenthen we may sayis continuous at

The sequential definition states that if for all sequenceswithwe havethenis continuous at

We must prove these statements are equivalent.

To do this, suppose a functionwith domainis either continuous or not at according to thedefinition.

Suppose first thatdoes satisfy thedefinition of continuity atWe want to deduce that ifis any sequence inwiththen

Suppose thatis given . By assumption there issuch thatfor allwithand there is an integersuch thatfor all

hencefor allso that

Next suppose thatdoes not satisfy thedefinition atWe want to find a sequenceinwithbut

By assumption there is somesuch that for eachfor some

Applying this statement with.we findfor some with

The sequenceis inwithbut