Math help, please see details?

Question

Skip this question if you are not at least college level math. I am using (sub i) meaning subscript is i.
How can I show the term:
[(summation from i=1 to n) h(sub i)]*[(summation from j=1 to m) k(sub j)]
is less than
n^2 {[h(sub1)]^2 + [h(sub 2)]^2 + ..... +[h(sub n)]^2 + [k(sub 1)]^2 + .... + [k(sub m)]^2}
For h(sub i) and k(sub j) being real numbers.
Do not show one example of it being true!!!!!!!
I need to show this for all real h,k.

Answer 1

Don't feel bad if you're not able to show that this holds for all real numbers: it doesn't. Consider:

n=1, m=8, h=[1], k=[1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2]

Then the first one is:
(1)*(1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2) = 4

But the second one is:
1² * (1 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4) = 3

4≮ 3

Therefore your conjecture is false.