WOLFRAM|DEMONSTRATIONS PROJECT

A Convergent Sequence Satisfies the Cauchy Criterion

choose limit a

1.

choose ϵ > 0

0.5

find M

1

i terms past M

1

j terms past M

2

zoom

1

This Demonstration shows that a convergent sequence satisfies the Cauchy criterion. Suppose

a=

lim

n∞

a

n

. For each

ϵ>0

, there exists

M∈

, such that for all

n>M

,

|a-

a

n

|<ϵ/2

. If

i,j>M

, then



a

j

-

a

i

=a-

a

i

+

a

j

-a≤a-

a

i

+

a

j

-a<ϵ

.