References ii
1. Problems on Sets,Sequences and Limits 1
1.1. Elementary Technique 1
1.2. Applications of the Stolz theorem 11
2. Inequality 14
3. Orders Estimate of In¯nitesimal 20
3.1. Notations and Examples 20
3.2. Exercises and homework 23
4. Application of In¯nitesimal: Wallis Formula and Stirling Formula 27
5. Topic on the Gamma functions 30
5.1. ¡ functions 30
5.2. Exercises 34
6. Working Technique in function theory 36
6.1. Iteration technique 36
6.2. Exercises and Homework 39
7. Applications of Di®erential 41
8. Treasures in Calculus 42
8.1. Euler's identi¯cation ³(2) = 1 + 1
22 ¢ ¢ ¢ + 1
n2 + ¢ ¢ ¢ ´ ¼2
6 42
8.2. Irrationality of ¼; log 2; ³(2); ³(3). 43
8.3. The properties of Tchebychev's functions 46
8.4. Mertens' Theorem and Selberg's Inequality 53
8.5. An Elementary Proof of The Prime Number Theorem by Selberg and ErdÄos 58
8.6. Gauss's Proof of The Fundamental Theorem of Algebra 59
9. Advanced Techniques in Analysis 60
9.1. Preliminary Results in Analysis 60
9.2. Scaling Technique and Schauder Estimate 72
9.3. Campanato's characterization of L2 functions to be HÄolder continuous 79
10. Advanced Topics in Analysis 82
10.1. Topic on Riemann Zeta Function(To be Continuous) 82
10.2. Elementary on Nevanlinna Theory(To be Continuous) 83
10.3. Elementary on p-adic Series(To be Continuous) 84

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