m] of Step 3 imply that (22) defines a Cauchy sequence
It is defined also the concept of f--statistically Cauchy sequence
and it is proved that if X is a complete space and [([x.
iii) If every Cauchy sequence
in X is convergent, then (X,d) is said to be a complete complex valued b-metric space.
A multiplicative metric space (X, m) is complete if every multiplicative Cauchy sequence
in it is multiplicative convergent to some x E X, .
A normed vector space V is called a Banach space if every Cauchy sequence
is a Cauchy sequence
in the Banach space C(I,R), hence converging uniformly to some x(.
Finally, S [subset] X is forward complete if every forward Cauchy sequence
is forward convergent.
is said to be strongly complete iff every strong Cauchy sequence
in V is strongly convergent to a point in V.
j=1] is a Cauchy sequence
of complex numbers for each fixed k [member of] [N.
By a complete non-Archimedean space we mean one in which every Cauchy sequence
m]} is a Cauchy sequence
and, thus has a limit point
Song  has pointed out there are some errors in the papers of Vasuki  and Grabiec , because definition of Cauchy sequence
given by Grabiec  is weaker than the one proposed by Song  and hence conditions of Vasuki's theorem and its corollary.