Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 25: Advanced Calculus
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Lecture(s) 
Sections 
Comments/Topics 

3 
1.1, 1.2 
Sets and functions. Logic and proofs. Proof by induction. 

1 
1.2, 8.4 
Algebraic and order axioms for the real numbers. Absolute values. 

1 
1.3 
Completeness axiom for the real numbers. Suprema and infima. 

2 
1.4 
Archimedian property of the real numbers. Density of the rational numbers. 

3 
2.1, 2.2 
Sequences. Definition of the limit of a sequence. 

3 
2.3 
Algebraic and order limit theorems. 

2 
2.4 
Monotone convergence. The limsup and liminf. 

2 
2.5 
Subsequences. BolzanoWeierstrass theorem. 

1 
2.6 
Cauchy sequences. 

3 
2.7 
Infinite series. Absolute convergence. Comparison test. Alternating series test. 

1 
2.8 
Double summations. Products of infinite series. 

3 
3.1, 3.2 
Topology of the real numbers. Open and closed sets. Accumulation, boundary, and interior points. 

2 
3.3 
Compact sets of real numbers. HeineBorel theorem. Finite intersection property. 

1 
3.4 
Connected and disconnected sets of real numbers. 
(2) A goal of this class is to ensure students learn to write rigorous proofs and communicate mathematical concepts using language. Have students regularly practice writing formal proofs that emphasize course content and mathematical thinking.