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There is plenty of math that is beautiful without needing much explanation of theory, such as fractals, geometric patterns and the Game of Life, that may interest beginners in mathematics. However, if they want to pursue the beauty of deeper forms of math, they inevitably encounter statements like:

Intermediate Value Theorem. If a function $f(x)$ is continuous over the interval $a \leq x \leq b$, then for every $f(a) < k < f(b)$ or $f(a) > k > f(b)$ there exists some $a < c < b$ such that $f(c) = k$.

Beginners might express discomfort when trying to digest such statements; to understand and work with them - which is necessary when pursuing math - they need to be able to

  1. Understand the precise meanings of mathematical statements, and form their own precise statements
  2. Manipulate such statements, such as finding contrapositives, inconsistencies, special cases or generalizations
  3. Hold several ideas in their head simultaneously, while keeping track of their truth-values and relationships between each other (which statements are justified by which others? Do any contradict each other?). Necessary in statements with many instances of "for all" and "there exist".
  4. Understand and form long chains of logical manipulations while quickly identifying and fixing mistakes
  5. Justify or find counterexamples to their own guesses

This list is probably not exhaustive, already it seems like a lot of practice is necessary to cultivate these abilities. That leads to my question:

Q: Are there good books, exercises or courses that can guide a person and help him/her acquire these skills through practice?

One potential obstacle is that one who gets confused by doing logical exercises may need a human teacher to clarify one's doubts...

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I've heard good things about both How to prove it and How to think like a mathematician. – goblin Jan 20 '14 at 5:32

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