Let ((A, B)) be a proximally complete pair in a metric space X and (A_{0}neqemptyset).
Let ((A, B)) be a cyclically complete pair in a metric space X and (A_{0}neqemptyset).
Since A and B are closed subsets of the complete metric space (mathbb {R}) and (d(A, B) = 0), it follows that ((A, B)) is a proximally complete pair.
end{cases} It is easy to see that ({z_{n}}) is a cyclically Cauchy sequence in (Acup B), and lim_{m,ntoinfty} {d(Tx_{n},Ty_{m})}=d(A,B). (34) But, by assumption, ((A, B)) is a cyclically complete pair in X so that either ({Tx_{n}}) or ({Ty_{m}}) converges.
Moreover, lim_{m,ntoinfty} {d(x_{n},y_{m})}=d(A,B). (16) Since by our assumption, ((A, B)) is a cyclically complete pair in X, either ({x_{n}}) or ({y_{m}}) converges.
But by assumption, ((A, B)) is a proximally complete pair in X, so that both sequences ({Tx_{n}}) and ({Ty_{m}}) have convergent subsequences in B and A, respectively.
By our assumption, ((A, B)) is a proximally complete pair in X so that the sequences ({x_{n}}) and ({y_{m}}) have convergent subsequences in A and B, respectively, that is, there exist subsequences ({x_{n_{r}}}) and ({y_{m_{s}}}) of ({x_{n}}) and ({y_{m}}) and ((x,y in(Atimes B)) such that lim_{rtoinfty}{x_{n_{r}}}=x quadtext{and}quad lim _{stoinfty}{y_{m_{s}}}=y.
But evidence from a new and more complete pair of autopsies and three months of intensive investigation by India's top police agency have added surprising power to arguments that the women did in fact drown, as the Kashmiri authorities initially announced, in a normally placid stream swollen dangerously with glacial runoff.
As defined by Rosenbaum (1989), "A complete matched pair is a set (mathfrak {I}) of N disjoint matched pairs, that is N matched pairs in which each treated unit appears once, and each control unit appears either once or not at all" (p. 1024).
