Would anyone be able to tell me how to prove that the orthogonal group over a local field for an anisotropic quadratic form is compact?

I am lazy, so I'll write this out as a sequence of claims without proofs. Let $K$ be the local field and let $ \ $ denote the absolute value on $K$. Let $V$ be the vector space with anisotropic form $\langle \ , \ \rangle$. Choose an arbitrary basis $e_1$, ..., $e_n$ of $V$. Define functions $ \ _{\infty}$ and $ \ _2$ from $V \to \mathbb{R}$ as follows: $$\left \sum a_i e_i \right_{\infty} = \max(a_i).$$ $$ v _2 =  \langle v,v \rangle^{1/2}.$$ Claim: The unit $\infty$ball, $B_{\infty}:=\{ v \in V: v_{\infty} \leq 1 \}$, is compact. Claim: The unit $\infty$sphere, $S_{\infty} := \{ v \in V: v_{\infty} = 1 \}$, is compact. Claim: The function $ \ _2$ is continuous. Claim: There is a positive constant $r>0$ such that $v_2 \geq r$ on $S_{\infty}$. (This is the step that uses anisotropy.) Claim: Define the unit $2$sphere by $S_2 := \{ v \in V: v_{2} = 1 \}$. Then $S_2 \subset (1/r) B_{\infty}$, and $S_2$ is compact. Claim: The orthogonal group embeds as a closed subspace of $S_2^n$, and is hence compact. 


To address Jim Humphreys' comment concerning a unified argument covering all (nonarchimedean) local fields, see the proof by Gopal Prasad (in "An elementary proof of a theorem of BruhatTitsRousseau and of a theorem of Tits", Bull. SMF 110) that a connected reductive group $G$ over a henselian nontrivially valued field $k$ has $G(k)$ bounded (equiv. compact, when $k$ is locally compact) if and only if $G$ is $k$anisotropic. Relevance: the special orthogonal group of an anisotropic nondegenerate quadratic form over a field $K$ is $K$anisotropic as a connected semisimple algebraic group over $K$ (so the question posed falls into the context of Prasad's argument). Indeed, arguing by contradiction, suppose there is a nontrivial split $K$torus in the special orthogonal group. This leads to a nontrivial $K$rational zero of the quadratic form by considering the weight space decomposition for the action of such a split torus on the $K$vector space in question (any nontrivial element in a single weight space is such a $K$rational zero). 


At least in the case of an algebraic extension $K$ of a $\mathbf Q_p$ with ring of integers $A$, you can also easily check (or read in O'Meara) that if a form $q$ on a vector space $V$ over $K$ is anisotropic, then the set of vectors $x$ with $q(x)\in A$ is an $A$lattive $L$ on $V$. Thus $O(V,q)=O(L,q\vert_L)$ is compact. 


The proof for the archimedean case is given in our own Pete Clark's notes., see Theorem 1. 

