https://www.amazon.com/dp/007154948X/?tag=pfamazon01-20 is a lifesaver, as was the first chapter of
Zorich Analysis I as well as the first chapter of Serge Lang's
Introduction to Linear Algebra. Another phenomenal book is
https://www.amazon.com/dp/0394015592/?tag=pfamazon01-20
Basically the greatest discovery I've had this year was to realize how proofs follow from
logic, i.e. implications, a chain of implications, logical equivalences etc... I think the first few
chapters of the Krantz book will give you the idea then you should do some further
research into how to use these ideas, another book that uses these ideas very well is
https://www.amazon.com/dp/0131481010/?tag=pfamazon01-20. The most preferable thing, to me, would be to do
Krantz & Bellman along with the first chapter of Lang, then the first few chapters of Lay.
Finish it off by doing the first chapter of Zorich & you'll be where I am now, trying to get
better
Before all of this I was extremely insecure about proofs, struggling to
understand the "logic" behind any of it & struggling to find patterns but now it's an
enjoyable experience of turning authors "wordy" proofs into a chain of logical implications,
well - assuming they are not too advanced or just incomprehensible to me :shy:
Note: These are personal preferences from experience, honestly all you need is Krantz,
Bellman & Lay as each gives insight I have not found in a single other book after
mercilessly searching,. The Lang chapter is just so beautiful as so much of the chapter is
derivable from a single chain of logic:
1) Take two vectors A & B 2) Make B longer than A (see page 23) 3) Find a vector orthogonal to B. 4) (A - cB)•B = 0 5) Use Pythagorean Theorem 6) ||A||² = (||A - cB||)² + ||cB||²
7) Prove that ||cB|| = |c|||B|| 8) ||cB||² = (√(cB)-(cB))² = c²B-B = c²||B||² 9) ||cB||² = c²||B||² ⇒||cB|| = |c|||B|| 10) ||A||² = (||A - cB||)² + |c|²||B||²
11) Notice c²||B||² ≤ ||A||² 12) Derive c 13) (A - cB)•B = 0 14) A•B - cB•B = 0 15) A•B = cB•B 16) c = (A•B)/(B•B) 17) c²||B||² ≤ ||A||² → 18) [(A•B)/(B•B)]² ||B||² ≤ ||A||²
19) [(A•B)/||B||²]² ||B||² ≤ ||A||² 20) [(A•B)²/||B||²] ≤ ||A||² 21) (A•B)²≤ ||A||²||B||² 22) A•B ≤ ||A||||B|| 23) C•C ≤ ||C||||C|| 24) Derive the Triangle Inequality Yourself!
(Sig on another forum 
).
Another list of books worth researching, as regards getting used to proofs, are:
Gleason - Fundamentals of Abstract Analysis
Maddox - Transition to Abstract Mathematics
Morash - Bridge to Abstract Math
Epp - Discrete Mathematics
Grimaldi - Discrete Mathematics
These looked like the best choices to me, hope this helps somewhat!
