markhkim

You can't have calculus without the real numbers, and you can't have asymptotic analysis and probability theory without calculus. There goes the whole field of the analysis of algorithms.

Nobody teaches the same topic in the exactly the same way.

There is, for one, far more material in any given subject than an introductory course or textbook can (or should) cover, so the author/instructor must choose what to include.

Plus, the order of presentation matters. For example, here are two standard ways of introducing the real numbers:

#1 (Dedekind cut). Picture a square of side length 1. The length of the diagonal, √2, cannot be represented by a ratio of integers, so we need a new number system to represent it. These numbers "in between" rational numbers are called irrational numbers, and together they form the real numbers.

#2 (Cauchy completion). Non-repeating decimals, such as π ≈ 3.141592, cannot be represented by a ratio of integers. We call such decimal numbers irrational numbers. Any number representable by a (finite or infinite) decimal is called a real number.

You can deduce #2 from #1, and vice versa. It's entirely up to the author/instructor to decide which one to start with.

Lastly, there is always a better way to explain the same material.

This was part of a collection of four essays on mathematical writing, commissioned by the American Mathematical Society:

  The committee was authorized by the Council of the American Mathematical Society in August 1968; the last appointment to it was made by Oscar Zariski, then president, in March 1969. The charge was to prepare "a pamphlet on expository writing of books and papers at the research level and at the level of graduate texts."
  
  In May 1969, two months after the committee was completed, one of its members resigned. He said he thought the project was too interesting to leave to a committee, which would never get it done properly, and he said he wanted to be free to write and publish his version independently. Norman Steenrod (the chairman) declined to accept the resignation, preferring to allow the member the freedom he sought. This left the exact membership of the committee up in the air.
  
  The work of the committee proceeded mainly on Steenrod's steam; he wrote to the other members (in triplicate), and occasionally they would write an answer (to him alone). The committee met only once (for an hour, at the Eugene meeting in August 1969, with three present). The result of the correspondence and the meeting was the decision to present to the Council, as the product of the committee, four separate essays, one by each of the four members, with the recommendation that the Society publish them, together, as this book. > > A year later (in August 1970) Steenrod had at hand only one essay. A year and six months later (in March 1971) that essay was published. (L'Enseignement Mathématique, 16 (1970), 123-152.) Even so, Steenrod was still hoping; he set August 300, 1971 as a target date for the receipt of all the essays. The solution he proposed for the problem created by the already published essay was to reprint it as is, as part of the AMS publication, provided the editors and publishers of L'Enseignement Mathématique agreed. They did.
  
  Steenrod died in October 1971, before quite completing his own essay. Before he died he asked, through his wife, that his nearly finished work be prepared for submission to the council and presented together with the others. That was done.
  
  Respectfully submitted,
  J. A. Dieudonné
  P. R. Halmos
  M. M. Schiffer

The other three essays are excellent as well and I recommend that you check out all of them. All of the authors are excellent mathematical expositors, though perhaps not as well-known as Halmos outside of the academic mathematics community. Dieudonné in particular had quite an illustrious writing career as well, having been part of the Bourbaki group as well as Grothendieck's EGA project.

I'm surprised no one mentioned this yet: make sure to talk to your professor, frequently. Introductory analysis courses exist primarily to teach a certain way of thinking, and there is, after all, no better way to learn how to think than to talk to someone who already knows the ways. Take advantage of what you have.