I'm trying to really solidify my maths knowledge so that I'm completely comfortable understanding why and how certain branches of mathematics are introduced in physics and inevitably that leads me to a study of proofs. I usually skip proofs as I found them annoying, unintuitive and just seemingly tedious guess work. I started a course on analysis at uni for my physics degree but switched option because I hated it so much. For example The Schwarz Inequality proof in a book I'm using: From which point it is clear the inequality can be proven. For me however, I immediately get stuck on where [itex]|c\rangle[/itex] came from. I tried to prove this on my own multiple times and probably spent about 6 hours total on it on and off and end up messing about writing [itex]|a\rangle[/itex] and [itex]|b\rangle[/itex] in terms of orthornormal vectors before feeling like an idiot and ultimately getting nowhere and just giving up. If left to work on this problem for a year i can't even imagine what on earth would cause me to think of introducing a third vector [itex]|c\rangle[/itex] into the mix. Then, not only that, but assigning it the very specific value: [itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex] I'd have a better chance of winning the lotto than happening to come up with this. Clearly I'm lacking something fundamental here, I'm guessing the choice of the vector c might seem obvious to someone good at this, but in my mind the great majority of non trivial proofs proceed in the following way: -> 1) Algebra that I can follow clearly -> 2) Insert step that seems to have been conjured out of thin air (ie lets introduce the vector [itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex]) -> 3) Algebra that I can follow clearly -> 4) QED. I then sit there for ages trying to justify how I would come up with step 2 and, i sort of can see why, but it is just 99.9% hindsight because once it comes time for me to do practise proofs on my own, I still don't even know where to start, and when i look for the proofs online, the above scenario plays out again. This repeats until I just eventually get fed up! This isn't my first foray into trying to learn these things, but I never get further than this and end up just deciding to just accept proofs and move on because its just a massive waste of my time and has seemingly little if any benefit.