Exercise from Terence Tao In the exercise below I don't understand why Exercise 13.5.9 doesn't contradict 13.5.8 ?

Exercise 13.5.9 is wrong as stated.  A topological space in which every sequence has a convergent subsequence is said to be sequentially compact.  Not every compact space is sequentially compact.
EDIT: Exercise 13.5.8 is also wrong.  This is not compact!  Let $S = \{x_i: i \in \mathbb N\}$ be a sequence of distinct points of $X \backslash \{\infty\}$, and consider the open cover $\{U_i: i \in \mathbb N\}$ where $U_i = S^c \cup \{i\}$.  This has no finite subcover.
