Existence of solution to minimization problems

A set if c y is called weakly closed if the conditions Un u weakly in V, Un G K, imply u G K. The following statement is valid. 

Theorem 1.10. A closed convex set of a reflexive Banach space is weakly closed. 

Now we can prove the statement of solution existence to minimization problems. 

Theorem 1.11. Let V be a reflexive Banach space, and K c V be a closed convex set. Assume that J V R is a coercive and weakly lower semicontinuous functional. Then the problem 

Let us take a minimizing sequence i.e. a sequence possessing the property 

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