Some exercies and homework I have during this Mathematical Analysis graduate course towards the Qual exam of PhD degree. I will upload the exercies and something I find interesting on this pages.

The solution and proof in some problems are not perfect, and welcome any disscusion and correction. And for Solutions to Walter Rudin's Principles of Mathematical Analysis, I am trying to do the problem in Rudin's book and I will try to update it sometime, but I am not sure if I can do it all.

• Logic
• Set Theory
• Cartesian products and functions
• Mathematical induction
• Relations, integers and real numbers

• Sequences and limits
• Exponential and logarithmic functions
• Subsequences, series and the Cauchy condition

• Subsequences, series and the Cauchy condition
• Limits and continuity

• Derivative
• Integral

• Metric spaces

• Uniformly Continuity
  • Stone-Weieratrass Theorem
• Banach Space
  • The Banach Contraction Principle
  • Arzelà-Ascoli Theorem

• Differentiability
• Higher Order Derivatives

• Higher Order Derivatives
• Local Maximum, Local Minimum
• Sylvester's Criterion