The purpose of this course is twofold. First, to give a careful treatment of calculus from first principles. In first-year calculus we learn methods for solving specific problems. We focus on how to use these methods more than why they work. To pave the way for further studies in pure and applied mathematics we need to deepen our understanding of why, as opposed to how, calculus works. This won't be a simple rehashing of first-year calculus at all. Calculus done this way is called real analysis.
In particular, we will consider what it is about the real numbers that makes calculus work. Why can't we make do with the rationals? We will identify the key property of the real numbers, called completeness, that distinguishes them from the rationals and permeates all of mathematical analysis. Completeness will be our main theme through the whole course.
The second goal of the course is to practise reading and writing mathematical proofs. The course is proof-oriented throughout, not to encourage pedantry, but because proof is the only way that mathematical truth can be known with certainty. Mathematical knowledge is accumulated through long chains of reasoning. We can't rely on this knowledge unless we're sure that every link in the chain is sound. In many future endeavours, you will find that being able to construct and communicate solid arguments is a very useful skill.
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