1. verify

let me know if the following is true

(a + b)^2 <= a^2 + 2ab + b^2

is there a name for this?

2. Its called a proof, expand the left hand side.

$
(a + b)^2 = (a + b) (a + b)= \dots
$

3. i know it's used to prove the cauchy schwarz inequality, but it seems like it's just ripped out of nowhere in that proof, i was just wondering if it's a definition or just a known equivalence etc

4. If you don't know how to do this, how can you possibly use the Cauchy-Schwarz inequality ?
The inequality you stated is in fact an equality... How did it appear in CSI's proof ?
Because CSI sometimes deals with multidimensional spaces and you may have misunderstood something (still judging by your question about f(x,y)

5. Originally Posted by Noxide
let me know if the following is true

(a + b)^2 <= a^2 + 2ab + b^2

is there a name for this?
It's true but strangely written! There is no need for the "<". For all a and b, $(a+ b)^2= a^2+ 2ab+ b^2$. I thought everyone learned that in secondary school if not before! If pressed to give it a name, I would just say "a perfect square".

6. Oh heh, I was/am quite aware of the identity of (a + b)^2
I just never saw it written with an inequality sign and wondered if it exists as an identity with the inequality sign or if it was just used in the context of the CSI proof.