Is $\omega_1 + 1$ sequentially compact? I believe there is a mistake in Terence Tao's Analysis II Exercise 2.5.8. It says 

Show that there exists an uncountable well-ordered set $\omega_1+1$ that has a maximal element $\infty$, and such that the initial segments $\{x\in ω_1+1:x<y\}$ are countable for all $y\in \omega_1+1\setminus \{\infty\}$. (Hint: Well-order the real numbers using Exercise 8.5.19, take the union of all the countable initial segments, and then adjoin a maximal element $\infty$.) If we give  the order topology (Exercise 13.5.5), show that $\omega_1+1$ is compact; however, show that not every sequence has a convergent subsequence.

I do not see how is the last assertion true. Every sequence in $\omega_1+1$ is bounded, hence it has $\lim \sup$, which is a limit point.
Can someone point me out to where I am wrong?
EDIT: This mistake has already been included in errata for the corrected 3rd edition. 
 A: $\omega_1+1$ is definitely sequentially compact: every sequence $x_n$ in it has either infinitely many times $\omega_1$ (the maximum) as a term, and we have our convergent subsequence, or we have infinitely many terms that are $< \omega_1$ and these have a supremum $x$ that is also $< \omega_1$, and then that in that subsequence we have subsequence that converges to that sup by definition of the order topology. The space $\omega_1 + 1$ is indeed compact, while $\omega_1$ is not. The max makes all the difference.. (For an ordered topological space $X$, $X$ is compact iff all subsets have a maximum, in fact.)
$\omega_1$ is even already sequentially compact (even easier, as we don't have a case of terms $\omega_1$ to consider). What is true, is that $\infty$, or really $\omega_1 \in \omega_1+1$ is in the closure of $\omega_1$ but not in its sequential closure; Tao might have been confused with that. Also, $\omega_1 \subseteq \omega_1+1$ is a sequentially compact subspace of a Hausdorff (even normal) space that is not closed. So the space does serve as a standard example of showing "sequences are not enough" in general topology.
