Statistical quasi-Cauchy sequences

A subset EE of a metric space (X,d)(X,d) is totally bounded if and only if any sequence of points in EE has a Cauchy subsequence. We call a sequence (xn)(xn) statistically quasi-Cauchy if st−limn→∞d(xn+1,xn)=0st−limn→∞d(xn+1,xn)=0, and lacunary statistically quasi-Cauchy if Sθ−limn→∞d(xn+1,xn)=0Sθ−limn→∞d(xn+1,xn)=0. We prove that a subset EE of a metric space is totally bounded if and only if any sequence of points in EE has a subsequence which is any type of the following: statistically quasi-Cauchy, lacunary statistically quasi-Cauchy, quasi-Cauchy, and slowly oscillating. It turns out that a function defined on a connected subset EE of a metric space is uniformly continuous if and only if it preserves either quasi-Cauchy sequences or slowly oscillating sequences of points in EE.