In many proofs, analogies to the real world aren't discussed or mentioned presumably because the audience and writers are non-novices and can think abstractly in terms of purely mathematical terms or maybe because analogies are risky because they help one memorize the proof rather than understand it. I haven't reached that level where I can do proofs without thinking in terms of analogies and I can't come up with analogies to make proofs myself.
To prove Cantor-Schorder-Bernstein theorem, I heard that one can use the analogy of Hilbert's Hotel to construct and apply functions repeatedly which is akin to the analogy of accommodating an infinite number of new guests arriving at an infinitely large hotel by endlessly giving them rooms and mapping the guests already in the hotel to new rooms via functions applied repeatedly or think about pouring water into to cups that are of different shapes yet to be proven are actually of the same size. Infectivity can be shown because every person is assigned to their own room with no two people having to occupy the same room. Subjectivity can be shown because everyone is assigned a room.
Source on Hilbert hotel and accommodating guests to prove CBS theorem:https://www.youtube.com/watch?v=IkoKttTDuxE
Source on pouring water analogy for CBS theorem:https://www.youtube.com/watch?v=_e7shdQWowQ
I find it useful to study proofs where analogies such as Hilbert's Hotel are possible because I can't think about the cantor-schroder-bernstein theorem or proofs in general in terms of simply math itself, using only the vocabulary of functions, injection, surjection.
My question is how do I find a list of proofs where analogies are possible to be made for proving the theorem, or are analogies always possible for proofs? I'm guessing these tend to be the elementary ones or set theory ones. I think I should practice with these first to become better at proofs.
Second, how do I gain the intuition behind proving things so I can come up with my own analogies to help me prove things or write proofs and eventually transition to not needing analogies like Hilbert's Hotel to prove the CBS (Cantor-Schorder Bernstein) and be able to go straight to doing function composition to prove the bijection for CBS theorem?
I've tried proving by examples and counterexamples, but I prefer to think in terms of analogies such as Hilbert's Hotel instead.