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While attempting Rudin's Principles of Mathematical Analysis, I only got about as far as page 9 before losing him in the proof that ##\mathbb{Q}## is dense in ##\mathbb{R}##. While his proof is only a few lines long, it does reveal some important properties that result from this theorem alongside the archimedean property which I find very insightful.
However, I needed a very long, in-depth, intricate explanation of the proof explained to me because I could not follow it, and was not able to derive the apparent corollaries that this proof revealed. Rudin used certain facts that I would not have been able to think of on my own no matter how many times I re-read his proof.
In an "easier" analysis book, it introduces these corollaries first, and then proves the density theorem afterwards (while Rudin does not mention these corollaries at all explicitly, so I assume he wants the reader to discover them on their own).
While I would prefer a presentation that includes excruciating details of proofs while assuming the reader has absolutely no abstract mathematical reasoning skill at all (or is not a very bright student in mathematics), I think too much hand-holding may be detrimental to math education so I am looking for a textbook that is somewhere in between the extremes of, say, Stewart's Calculus and Rudin's PMA (while one expects a bit much from the reader, the other assumes too little of the reader's knowledge, I guess).
Do you guys have any suggestions for introductory analysis textbooks for students that do not pick up concepts too quickly (something around the ballpark of at least being able to understand 9 pages of the textbook in less than 1.5 weeks, which is approximately how long I spent on the first 9 pages of Rudin and still didn't understand that particular proof).
However, I needed a very long, in-depth, intricate explanation of the proof explained to me because I could not follow it, and was not able to derive the apparent corollaries that this proof revealed. Rudin used certain facts that I would not have been able to think of on my own no matter how many times I re-read his proof.
In an "easier" analysis book, it introduces these corollaries first, and then proves the density theorem afterwards (while Rudin does not mention these corollaries at all explicitly, so I assume he wants the reader to discover them on their own).
While I would prefer a presentation that includes excruciating details of proofs while assuming the reader has absolutely no abstract mathematical reasoning skill at all (or is not a very bright student in mathematics), I think too much hand-holding may be detrimental to math education so I am looking for a textbook that is somewhere in between the extremes of, say, Stewart's Calculus and Rudin's PMA (while one expects a bit much from the reader, the other assumes too little of the reader's knowledge, I guess).
Do you guys have any suggestions for introductory analysis textbooks for students that do not pick up concepts too quickly (something around the ballpark of at least being able to understand 9 pages of the textbook in less than 1.5 weeks, which is approximately how long I spent on the first 9 pages of Rudin and still didn't understand that particular proof).