MAT 420 - Real Analysis I

1. Describe the important differences between supremum and maximum, and between infimum and minimum and prove statements involving suprema and/or infima of sets.
2. Give the formal definitions of concepts such as convergence, limit, and derivative and explain them.
3. Apply formal definitions to prove statements about convergence, limits, and derivatives.
4. Use limit laws to determine sequence convergence, limit values, and function limits.
5. Articulate with examples and counterexamples, the relationships between the various properties of sequences and functions.
6. Explain the Intermediate Value Theorem, Extreme Value Theorem, the Mean Value Theorem, the first derivative test, and l'Hôpital's Rule and their reliance on continuity or differentiability.