In the first part of this course, we covered the construction of real number, and some results about limit. Here is a list of key concepts

• The numbers $\N, \Z, \Q$.
• The axioms of field, an example of finite field $\F_5$.
• The order relation, and ordered set. upper bound, lower bound. The least upper bound property.
• $\R$ as equivalence classes of Cauchy sequences in $\Q$. Prove many familiar operations and properties of $\R$.
• $\R$ has least upper bound property. (hence $\sup$ and $\inf$ of bounded subset in $\R$ exists in $\R$)
• Sequences in $\R$, notion of convergence
• Monotone bounded sequences are convergent (for increasing sequence, the $\lim a_n = \sup\{a_n: n \in \N \}$; for decreasing one, $\lim a_n = \inf \{a_n: n \in \N\}$.
• lim-sup and lim-inf. The “epsilon of room” philosophy.
• Cauchy sequences are convergent.