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Exact(25)
Since is bounded, we can choose a subsequence of which converges weakly some point x.
Since is bounded, we can choose a subsequence of which converges weakly to some point x.
Since is bounded, we can choose a subsequence of converging weakly to.
Since the sequences are bounded, we can choose a subsequence of such that for each.
Since is bounded, we can choose a subsequence of such that (3.33).
To prove (3.22), since is bounded, we can choose a subsequence of which converges weakly to and (3.23).
Similar(35)
Since K is bounded, we can chose (varepsilonin 0,pi/2)) and (k>0) so that (operatorname{diam}(K leqfrac{pi/2-varepsilon}{sqrt{K leqfrac{pi/2-varepsilon
Since are bounded, we can always choose a positive vector such that (322).
Since is bounded on, we can choose such that on, and so that (3.52) takes the form (3.53).
In other words, for this bounded, we can write:.
Since is bounded, we always can choose a sufficiently large such that (311).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com