let me know if the following is true
(a + b)^2 <= a^2 + 2ab + b^2
is there a name for this?
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)