From looking at the images you posted it seems that you are proving:
The standard proof follows from the fact where is the angle between the two non-zero vectors .
Then note that
first of all, you have and wrong.
while:
, as well.
thus:
.
on the other side of the inequality we have:
since , we indeed have for these particular U,V:
.
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for your second problem, you are also calculating the norms of U and V incorrectly. use the definition of the inner product you are given!
still incorrect. for starters, you have an extra 2^{-n} term in and .
secondly, you don't seem to get that the inner product of two vectors should be a SCALAR, not an "expression".
so for U, you should have:
and for V, you should have:
while for the dot product, you should have: