For odd primes $p$, the Sylow $p$-subgroups of Diherdral group are cyclic and normal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.5
Solution: Let $P \leq D_{2n}$ be a Sylow $p$-subgroup, where $p$ is an odd prime. Note that every element of the form $sr^a$ has order 2 and thus by Lagrange’s Theorem cannot be an element of $P$; thus $P \leq \langle r \rangle$. Hence $P$ is cyclic. Moreover, in Exercise 4.4.9 we showed that every subgroup of $\langle r \rangle$ (in particular $P$) is normal in $D_{2n}$.