My experience in learning to do proofs was transformed when I read the first chapter of Principles of Mathematics, by Allendoerfer and Oakley, simply because it laid out the basic facts of the "propositional calculus". Namely what does it mean to prove something? I.e. when do you know a statement is true? A fundamental fact is the "contrapositive" principle, namely that to prove that A implies B, it is entirely equivalent to prove that B being false implies A is false also.
If you don't know this, you are lost when some book tries to prove that every continuous function on [0,1] is bounded, and they do so by starting out assuming the function is not bounded and concluding it is not continuous. Then they stop, and you go, .. ok, huhhh?
The book I mentioned is hard to find at a good price, but any book that explains the propositional calculus, namely what does it mean to say a given statement is true, will do.
(The following book may be essentially the same, but I am not positive:
https://www.amazon.com/gp/product/B0000CKBXJ/?tag=pfamazon01-20)
the most famous principle is "modus ponens", namely if A is true and A implies B then B is also true. The classic example is "All men are mortal, Socrates is a man, hence Socrates is mortal", which is actually a bit more complicated since it also involves the "quantifier" word "all".
You need to learn that proving both A and B are true, is the same as proving neither one can be false.
Similarly, proving that at least one of A or B is true, is the same as proving they cannot both be false. Equivalently, if A is false, then B must be true.
Similarly, to prove that A and B cannot both be true is the same as proving that at least one is false. Equivalently, if A is true, then B must be false.
This is the game you must learn to play, which requires first learning very well just what it actually means to say a given statement is tue, and what statements are equivalent to each other in terms of truth value.
To repeat, before you can prove a given statement is true, you must know exactly what it means to say it is true, and in particular you must know what other statements are equivalent to that statement.
You must also know how to phrase a statement that is the "opposite" of the given statement, namely the "negation" of a given statement. This is because proving a given statement is true, is often done by proving its negation cannot be false. YUou cannot do this if you do not know how to form the negatino of a statement. E.g. one of my exam questions was "find the negation of the statement: 'mathwonk is both stupid and lazy'". The answer is "mathwonk is either not stupid or not lazy", or perhaps: "even if mathwonk is stupid, at least he is not lazy".
To prove e.g. that either A or B is true, one often proceeds by assuming that A and B are both false, and concluding that some other statement C would then also have to be true, and then noting that C is actually false. Then since the falsity of both A and B forces the truth of an impossibly false statement C, one concludes that A and B cannot both be false, hence at least one must be true.
This is based on the principle that if assuming the truth of A implies the truth of B, but B is in fact clearly false, then A must also be false.
Another basic principle is that to prove A implies B, it suffices to show that A is in fact false. E.g. if A is "You will jump off the high board", and B = "I will jump off the high board", then A implies B is the statement "I will jump if you will". Then suppose that in fact you will not jump. Then whether I jump or not, the statement "if you jump, I will" is true. I.e. if you won't jump I don't have to jump in order to prove this statement. It is automatically true.
Note here that the fact you will not jump does not prove that I will jump, rather it proves only the conditional statement that "I will jump IF you jump".
A basic error of this type that people make is confusing the statement "if A is true then B is true", with the statement "B is true". For this conclusion, one needs also that A is in fact true.
E.g. people think that the derivative of fg is f'g + fg'. This is NOT necessarily true, unless both f and g are differentiable. I.e. the product rule does NOT say that (fg)' = f'g + fg', it says, IF f and g are both diffetentiable, THEN (fg)' = f'g + fg'.
So a basic lesson about proofs and theorems, is that all theorems have hypotheses, if the hypotheses are not true, then the conclusion may not be either.
This is another example of "modus ponens", i.e. if both statements "if A is true then B is true", and "A is true", are true, then the statement B is also true. But if only the statement "if A is true then so is B", is true, and we do not know about A, then we cannot deduce the truth of B.
Proofs can be tricky also because statements in ordinary English are often hard to interpret precisely in mathematically logical terms. Quantifiers are especially important, such as "for all" and "for some". The word "any" is particularly troublesome since in different contexts it can mean either one of those. In fact even the previous sentence is unclear since it can mean there are circumstances where "any" means "for all", and other circumstances where "any" means "for some", or it can mean there are circumstances where it could mean both. I intended to say there are circumstances where you cannot be sure which meaning to assign to "any".
Hence, in mathematics we need to be very careful in our choice of words, to reduce the number of possible interpretations. Still you have to learn the full range of standard interpretations in order to first understand, then prove, a mathematical statement.
