**Problem 1** Figure out how this website works! Make a login and introduce yourselves on the Class Notes page marked "Beginnings". The Lipschitz inequality is shown as: $f(a)-f(b) \leq l ||a-b||$.

**Problem 2** Suppose $[a,b]$ is a closed interval in $\mathcal{C}$. Suppose $B \subset \mathcal{C}$ has the property that both the first point of $B$ and the last point of $B$ exist and are contained in $[a,b]$. Show that $B$ is contained in $[a,b]$.
**Solution 2** Because the first point of B exsits and is contained in $[a,b]$, we know that the first point of B assumed as $\underline{B}$ satisfying $\underline{B} \geq a$. Similarly, we know that the last point of $B$ assumed as $\bar{B}$ satisfying $\bar{B} \leq b$. Thus, we can know that the set $[\underline{B},\bar{B}] \subseteq [a,b]$ and $B=\{ \underline{B}, ..., \bar{B}\}$ such that $B \subseteq [\underline{B},\bar{B}]$. Therefore, $B$ is contained in $[a,b]$.

**Problem 3** Suppose $c \in \mathrm{ext}(a,b)$, where

Show that $c$ is not a limit point of $(a,b)$.

**Problem 4** Suppose $A \subseteq \mathcal{C}$ is a set with no limit points. Show that for every $a \in A$, there exists an open interval $I$ such that $A \cap I = \{a\}$.

**Problem 5** Suppose $A \subseteq \mathcal{C}$ has the property that for every $a \in A$, there exists an open interval $I$ such that $A \cap I = \{a\}$. Does it follow that $A$ has no limit points? Explain your answer.

**Problem 6** Recall that $\mathbb{Z}$ with the usual $<$ relation is an endless ordered set. Show that no subset of $\mathbb{Z}$ has a limit point.