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Sequential Compactness

Please see the attached file for the fully formatted problems.

Consider C[0, 1], the space of real valued continuous functions defined on
the unit interval [0, 1]. Let
K = C1[0, 1] {f :
Z 1
0
f02
 1, ||f||1  1}
Note that C1[0, 1]  C[0, 1], and K  C[0, 1]. Show that K is compact in C.
I am assuming compactness here refers to the sequential compactness. This
seems to make the most sense. Since this problem is an analysis problem,
please be sure to be rigorous, and include as much detail as possible so that
I can understand. Please also state if you are making use of some fact or
theorem. Thanks!

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Solution Summary

Sequential compactness is proven. The solution is detailed and well presented.

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