Proposition 1.7.
Assuming that:
(
K
,
|
∙
|
)
is
non-archimedean
(
x
n
)
n
=
1
∞
a sequence in
K
|
x
n
−
x
n
+
1
|
→
0
Then
(
x
n
)
n
=
1
∞
is Cauchy. In particular, if
K
is in addition complete, then
(
x
n
)
n
=
1
∞
converges.