A simple matan for beginners?

Question

Mathematics

Soon I'm passing a test on the main theorems of the foundations of mathematical analysis (sequence stabilizability, sequence limit, function limit, continuous functions).
Please advise the most optimal and, if possible, simple textbook (reference book), where the main theorems on the topics listed above are proved. Many authors, in particular Krasnov, omit the proof of many important theorems.
I will be grateful))