# Key Topics from Undergraduate Analysis

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For those who would like a review of undergraduate analysis, here is a list of good textbooks and important topics, including links to the relevant wikipedia pages. The wikipedia pages are of mixed quality. For those who are interested, it would be an excellent exercise to create new pages on this wiki explaining the important topics. You may organize these however you see fit. You can request an account to edit the wiki using the link on the top right of this page. If it takes me too long to remember to approve your request, send me an email :).

## Undergraduate Analysis Textbook Recommendations

- Rudin, Principles of Mathematical Analysis
- Elementary Analysis by Kenneth Ross, second edition (using this link and your UCSB id, you can download the book for free)

## List of Topics

- Limit of a sequence, [1],
- Cauchy Sequence, [2]
- Subsequential limit, [3]
- Limit of a function, [4]
- Pointwise convergence, [5]
- Uniform convergence, [6]
- Continuous function, [7]
- Differentiable function, [8]
- Riemann integral, [9]
- Domain, [10]
- Monotone convergence theorem, [11]
- Heine–Borel theorem, [12]
- Bolzano–Weierstrass theorem, [13]
- Completeness of the real numbers, [14]
- Open set, [15]
- Neighbourhood, [16]
- Cantor set, [17]
- Completeness, [18]
- Limit superior and limit inferior, [19]
- Supremum, [20]
- Infimum, [21]
- Contraction mapping, [22]
- Triangle inequality, [23]
- Metric space, [24]
- Dense set, [25]
- Countable set, [26]