Show that the following is an inner product Currently, I've started to take Analysis 2 in school and we are doing Euclidean Spaces.  There is an example that I'm trying to prove but I can't wrap my mind around. 
$x, y \in \mathbb R^d$ 
$\langle x,y \rangle:= \sum^d_{j=1}x_j-y_j$
So, there are 4 rules for this to be an inner product. Non-negativity, definite, symmetric, linearity. But I think this isn't non-negative; am I wrong? $y$ vector can always be bigger than $x$ and that can make the sum negative? Or am I missing something?

There is a very easy way to see that this is not an inner product. Note that it often happens that inner products are negative for given choices of $x$ and $y$. However, what happens when you compute $\langle x,x\rangle$ for any $x$? Is this a positive number for all $x\neq 0$?
