In search for books like these in the past, I found "Understanding Analysis" (Stephen Abbott). I went through the first two chapters and I liked it. It is written in a narrative which is both entertaining and instructive. He explains the problem, why is it relevant, ways of approaching it, etc. "It is designed to capture the intellectual imagination."
From the preface:
"This book is an introductory text. The only prerequisite is a robust understand-
ing of the results from single-variable calculus. The theorems of linear algebra
are not needed, but the exposure to abstract arguments and proof writing that
usually comes with this course would be a valuable asset. Complex numbers are
never used.
The proofs in Understanding Analysis are written with the beginning student
firmly in mind. Brevity and other stylistic concerns are postponed in favor
of including a significant level of detail. Most proofs come with a generous
amount of discussion about the context of the argument. What should the
proof entail? Which definitions are relevant? What is the overall strategy?
Whenever there is a choice, efficiency is traded for an opportunity to reinforce
some previously learned technique. Especially familiar or predictable arguments
are often deferred to the exercises.
The search for recurring ideas exists at the proof-writing level and also on
the larger expository level. I have tried to give the course a narrative tone by
picking up on the unifying themes of approximation and the transition from the
finite to the infinite. Often when we ask a question in analysis the answer is
“sometimes.” Can the order of a double summation be exchanged? Is term-by-
term differentiation of an infinite series allowed? By focusing on this recurring
pattern, each successive topic builds on the intuition of the previous one. The
questions seem more natural, and a coherent story emerges from what might
otherwise appear as a long list of theorems and proofs."