Construct an expression that’s similar to (or part of) something in the premise, then directly relate it to another.
It’s often more helpful to use the premise at a later point in the proof, especially if it’s a constructive one. For example, you let there be an object, and then at some point you can use the premise to say something about the object.
A good proof by contradiction can’t be easily transformed into a direct proof. It uses the contradiction to derive an alternate reality with extra information, which eventually proves to be incorrect.
Don’t blindly expand the syntax of an expression; instead, focus on its semantics.
A corollary is that the equivalent of rubber duck debugging won’t get you anywhere.
Textbook proofs are often too condensed to be informative. Instead, derive the proof yourself.
Start with the belief that the claim is incorrect, then convince yourself that it’s correct.
Try iteratively refining a proof sketch until it’s a rigorous proof, like animation storyboarding. This helps in particular when you’re stuck, e.g., going from A to C may be easier than A to B to C, especially if B and C are unknown.
Real Analysis
Knowing every single definition and theorem is important, since the concepts are densely connected.
Statistics and Probability
It’s clean and helpful to manipulate the expressions coarsely rather than finely, e.g., working directly with the equalities within \(Pr(\cdot \leq \cdot)\).
Also proving some V-statistics estimators are biased relative to U-statistics.
Equivalent forms of distributions (and theorems linking them to some definitions) are often more useful (and simple) toward proving some properties like convergence in distribution.
E.g., Levy’s continuity Thm., MGF convergence.
A simple method to generalize proofs is to show some result on the real line and then use isometry to Borel sets.