I am having trouble applying this 'hint' to the original question. The original question was -
So I have shown that the hint is true but I can't see how I can just substitute in the series a_n and b_n for x and y as the |xy| cannot be represented by \sum_{n=1}^{\infty}a_nb_n as that would be...
All I can think of is that the square of a real number will be a positive finite real number..?
So it doesn't matter what is in the brackets it will always be greater than or equal to zero and therefore (|x|-|y|)^2 >=0 or basically (...)^2 >= 0 is true. Is that correct?
Ok, if I collect the terms on one side I get
|x|^2 -2|xy| + |y|^2 >= 0
which is
(|x| - |y|)^2 >= 0
So I can say
|x| - |y| >= 0
But this isn't true as you could have, for example, x = 5 and y = 7. Am I missing something here?
Homework Statement
This is part of a question on absolute convergence on series. The following equation is given as a hint. It says that before answering the question on series I should prove that |xy| <= 1/2(|x|^2 + |y|^2) for any x,y ε R
Homework Equations
The Attempt at a...