Let R be the real numbers,
S ⊂ R and E be an ordered topological vector space. Sufficient conditions are given
that a sequence {yk},yh: S → E, will have a subsequence {hk} such that for each
i ∈ S,{hk(t)} is either eventually monotone or else is convergent. In case E is a
Banach space, sufficient conditions are given that {yk} have a subsequence {hk} so
that {hk(t)} converges for each t ∈ S. Finally, if E = R, the concept of
{yk} being equioscillatory is defined and it is shown that a necessary and
sufficient condition for {yk} to have a subsequence that converges at every
point of S is that {yk} have a subsequence which is pointwise bounded and
equioscillatory. An application of these results to differential equations is treated
briefly.
