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
.