List of Proofs (Math 351)

(pg = page in the textbook,
Ex = Example, Th = Theorem, 
C = Corollary, L - Lemma,
Ec = Exercise, S = Section)

1 Various proofs by Mathematical induction
(as Ex 1, pg 3, Ex 2, pg 4, and many Exc in S 1)

2 Th 2.2 (Rational Zeros Theorem), pg 9 and its applications
as Ex 6, pg 11.

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3 Various proofs for limits of sequences using the definition
of the limit as Ex 1, pg 37, Ex 3-4 pg 38-40 and Ex 6 pg 50.

4 Formal proofs for limits of sequences as Ex 5-6 pg 40-41,
Th 9.1 pg 43, Exc. 9.12 pg 53. 

5 Arithmetic oprations with limits Th 9.4, L 9.5, pg 44-46.    

6 Infinite limits (proofs using the definition) Th 9.9-9.10
pg 50-52.

7 Monotone sequences Th 10.2, pg 55 and C 10.5, pg 57. 

8 limsup/liminf Th 10.7, pg 58-60, Th 12.1 pg 75-76.

9 Cauchy sequences L 10.9, pg 60, Th 10.11 pg 61.

10 Sequence of all rational numbers, Ex 3, pg 65.

11 Bolzano-Weierstrass Theorem and its prerequisites
Th 11.3 pg 67, C 11.4 pg 68-69, Th 11.5 pg 69.   

12 Set of subsequential limits
Th 11.7 pg 71, Ex 10, pg 72, Exc 11.9, pg 74.

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13 Infinite series Th 14.4 pg 93, C 14.5 pg 93 
Ex 4 pg 96-97, Ex 7 pg 97, Exc 14.8 pg 99.

14 Tests for convergence of series Th 14.6 pg 93-94, 
Th 14.8-14.9 pg 94-95, Th 15.1 pg 102.

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15 Continuous functions Th 17.2 pg 116, Ex 1 pg 117, 
Ex 3, pg 119, Th 17.4, pg 121-122

16 Function defined differently on rational and irrational 
numbers Exc 17.12 and 17.14, pg 125.

17 Min/Max and Intermediate Value Theorem 
Th 18.1-18.2 pg 126-128, Ex 1 pg 128, Exc 18.8. pg 131m
Exc 18.12, pg 132. 

18 Uniform continuity Ex 2-3, pg 134-135, Th 19.2 pg 136. 

19 Properties of uniformly continuous functions
Th 19.4, pg 138-139, Th 19.6 pg 140-142, exc. 19.4, pg 144.

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20 Limits of functions Ex 2, pg 147-148, Ex 4 pg 148-149,
Th 1-3 from the Hand-out.

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21 Power series Th 23.1, pg 172, Ex 4 pg 173, 
Exc 23.7-23.9, pg 176-177.

22 Uniform convergence of series Ex 5, pg 179-180, 
Th 24.3, pg 180-181, Ex 7, pg 181-182. 

23 Weierstrass M-test Th 25.2 pg 185, 
Th 25.7, pg 189 (not that we had a much simpler proof in class)

24 Differentiation and integration of power series
Th 26.1, pg 192, C 26.2, pg 192, L 26.3, pg 193, 
Th 26.4-26.5, pg 193-195, Exc 26.8, pg 200.

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25 Differentiation of functions Ex 3, pg 206-207, 
Th 28.2-28.3, pg 207-209. 

26 Mean Value Theorem Th 29.1-29.3 pg 214-215, C 29.7 pg 217.

27 Taylor's Theorem Th 31.3, pg 233-234, C 31.4 pg 233,
Ex 1-2, pg 233-234.

28 Binomial Series Theorem 31.7, p 236-238.