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Hence, there exists a subsequence, still denoted by ({v_{n}}).
and there exists a subsequence which is still denoted by such that.
The sequence has a weakly converging subsequence still denoted such that.
Thus, for a subsequence still denoted by there is such that (3.25).
Then as in the proof of Theorem 3.1 possesses a convergent subsequence (still denoted by ).
We now consider a functional from into still denoted by, by.
Again, passing to a subsequence of, still denoted by, we assume in addition that (3.23).
Passing to a subsequence of, still denoted by, we may assume that and exists.
From (3.44 and (3.57), we can extract a subsequence from, still denoted by, such that (3.59).
Then there exists a subsequence of, still denoted by which weakly converges to.
Hence, ((v_{n})_{n}) admits a weakly convergent subsequence, still denoted by ((v_{n})_{n}).
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com