We recommend that you read the lecture notes and the textbook concurrently with (or prior to) the lectures. If you only read these sources occasionally and after the fact (e.g. when your homework is due) then you will not get the most out of the course.

• Week 1: Introduction to analysis; the natural number system; induction; the integers; the rationals
• Week 2: Cauchy sequences; the real numbers; the least upper bound property; cardinality of sets
• Weeks 3/4: Countable and uncountable sets; sup, inf, limsup and liminf; limits and limit points
• Week 5: Exponentiation, standard limits, series, convergence tests, rearrangement of series
  
• Week 6: Subsequences; Bolzano-Weierstrass theorem; limiting values of functions; continuity; left and right limits
  
• Weeks 7/8: Maximum principle; Intermediate value theorem; Uniform continuity; Differentiability; Mean value theorem; Inverse function theorem
  
• Week 9: Partitions of intervals; piecewise constant functions; the Riemann integral; Riemann sums; properties of the Riemann integral
  
• Week 10: Riemann-Stieltjes integrals; the fundamental theorems of calculus; products and absolute values of Riemann integrable functions; integration by parts; change of variables formula