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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:

For any pair of vectors [itex]|a\rangle[/itex],[itex]|b\rangle[/itex] in an inner product space V, the schwarz inequality holds:

[itex]\langle a|a\rangle \langle b|b\rangle \geq |\langle a|b\rangle |^2[/itex]. Equality holds when [itex]|a\rangle[/itex] is proportional to [itex]|b\rangle[/itex]

The proof then starts:

Let [itex]|c\rangle = |b\rangle - (\langle a|b\rangle / \langle a|a\rangle)|a\rangle[/itex]

Therefore [itex]|b\rangle = (\langle a|b\rangle / \langle a|a\rangle)|a\rangle + |c\rangle[/itex]

Then take inner product of [itex]|b\rangle[/itex] with itself:

[itex]\langle b|b\rangle = \left|\dfrac{\langle a|b\rangle}{\langle a|a\rangle}\right|^2 \langle a|a\rangle + \langle c|c\rangle[/itex]

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.