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subsequence of {&}, . indexed by S2 ç; SI. Proceeding in a sirnikir fashion, we may
conclude that there exist subsequences indexed by S=& such that (w:}converges for every i=l.
7. ..., 1. when ES. The inequalities in the assumption hold for any j>k. so they hold in
particular for adjacent elements of the subsequence S. Iabeled ka. ka+ Because the difference
-Lm*/ _
c 2
is bounded below by O and above by an expression which converges to O through the
subsequence S. the daim is proven.
subsequence S = {k,, }:, E N such thar the objectives converge ro equal values in rhe limit. i.e.
Frorn~pi . we have
Subtracting (3.4) from (3.5) gives