How to improve the ability to problem solving in mathematical analysis?

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I recently learn Mathematical Analysis by Zorich and feel ok with theorems and examples in the book, however I'm stuck when do exercises in Problems in mathematical analysis by Kaczor, I refer to solutions and find that there are plenty of details that seem to come from vacuum, for example, to prove an inequality the author use a transformation of binomial formula and I don't understand what is the original idea of using such formula, how can one get to this detail from the beginning and step by step without consulting the solutions. Is there any suggestion? Thank you a lot.