Week 1: Induction

• Topics: Mathematical Induction.
• Key Problems:
  • Proving summation formulas: \(\sum i^3 = (\sum i)^2\).
  • Proving inequalities: \(n^2 > n+1\) for \(n \ge 2\), \(n! > n^2\) for \(n \ge 4\).
  • Proving divisibility/parity properties.

Week 2: Supremum, Infimum, and Sequences

• Topics: Infimum/Supremum, Sequence Convergence, Squeeze Lemma.
• Key Problems:
  • Finding inf/sup of sets (e.g., \(\{(-1)^n/n\}\)).
  • Proving limits using definitions.
  • Using Squeeze Lemma for limits like \(\lim (s_n t_n) = 0\) if \(s_n \to 0\) and \(t_n\) is bounded.
  • Rational/Irrational density examples.

Week 3: Limit Theorems and Recursive Sequences

• Topics: Limit arithmetic, Recursive sequences.
• Key Problems:
  • Calculating limits of rational functions (divide by highest power).
  • Recursive sequence limits: \(s_{n+1} = \sqrt{s_n + 1} \to \frac{1+\sqrt{5}}{2}\).
  • Limit properties: \(\lim (ks_n) = k \lim s_n\).

Week 4: Subsequences and Limsup/Liminf

• Topics: Monotone subsequences, Subsequential limits, Limsup/Liminf.
• Key Problems:
  • Finding monotone subsequences of \((-1)^n\), \(1/n\), etc.
  • Determining the set of subsequential limits.
  • Calculating \(\limsup\) and \(\liminf\).
  • Constructing sequences with specific limit properties (e.g., rationals enumeration).

Week 5: Series and Limsup Properties

• Topics: Series Convergence Tests, Limsup properties.
• Key Problems:
  • \(\limsup |s_n| = 0 \iff s_n \to 0\).
  • \(\limsup (s_n + t_n) \le \limsup s_n + \limsup t_n\).
  • Ratio Test applications: \(\sum 2^n/n!\) (converges), \(\sum n^2/3^n\) (converges).
  • Comparison Test: \(\sum 1/\log n\) (diverges).
  • Linearity of series sums.

Week 6 & 7: Continuity

• Topics: Continuity definitions, \(\epsilon-\delta\) proofs, Intermediate Value Theorem.
• Key Problems:
  • Proving continuity of composite functions (\(\log(1+\cos^4 x)\)).
  • \(\epsilon-\delta\) proofs for \(x^2\), \(\sqrt{x}\).
  • Continuity of "Dirichlet-like" functions (continuous only at 0).
  • IVT applications: Fixed points (\(x = \cos x\)), Roots of odd-degree polynomials.
  • Unbounded continuous functions on open intervals (\(1/x\) on \((0,1)\)).

Week 8: Uniform Continuity

• Topics: Uniform Continuity definitions and theorems.
• Key Problems:
  • Uniform continuity on closed bounded intervals (\(x^3\) on \([0,1]\)).
  • Non-uniform continuity of unbounded functions (\(1/x^3\) on \((0,1]\)).
  • \(\epsilon-\delta\) proofs for uniform continuity (\(3x+11\), \(x^2\) on \([0,3]\), \(1/x\) on \([1/2, \infty)\)).