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It is not clear whether the question is about a student or a student. On the one hand, matanalysis is mainly a university subject, on the other hand, the word “in-depth” is usually used in school discourse. I can recommend several books where mathematical analysis (or its sections) are presented especially clearly, so much so that the book is easy to read in the order of self-education.
Demidovich's problem book has already been mentioned. After reading the introduction to each chapter, you can solve quite a few problems from it.
There is an excellent, but rather peculiar book by Zeldovich (Ya. B. Zeldovich Higher Mathematics for beginners and its applications to physics). There are few formulas and many things are well explained at a qualitative level.
The book of Gelbaum and Olmsted (Gelbaum B., Olmsted J. Counterexamples in analysis) – – – an unsurpassed masterpiece and indispensable for getting into the subtleties of analysis. It's also brilliantly written.
I will name two classic textbooks that, in my opinion, are written better and clearer than others.
More concise — Rudin's textbook (U. Rudin). Fundamentals of mathematical analysis.) This book is the quintessence of analysis, a kind of digest, all the most important things. It's also beautifully written.
More verbose – – – two-volume book by Zorich (Zorich V. A. Mathematical analysis. Part 1, Part 2). It “has everything”, but it is written in a very human language, easy to read and pleasant.
(Of course, all these books are available online; for example, you can download them from techlibrary.ru and a lot more from where.)
My favorite site – //coursera.org To date, I have found 82 courses from the world's best universities in different languages (most of them are in English, but there are others, including 2 courses from the Higher School of Economics in Russian).
Portal edx.org traditionally, it offers imho courses with a more complex level and is now available only in English, there is a mat for various obalastyam. All 123 courses have been opened.
And from the manuals I can also recommend the classics of the genre-the author beloved by all students-B. P. Demidovich. Read the explanation and solve problems. If you have the willpower and brain, then you can master the course yourself, which is designed for about 2 years of study at the initial courses of the university.
Then you can look at A. N. Kolmogorov.