analytic vector bundles Let $E$ be a real analytic vector bundle on an analytic manifold $M$. Assume that $E$, as  a smooth vector bundle, is  a trivial bundle.

Is $E$ a trivial analytic vector bundle?

I need to the answer to this question for the following question:
Analytic version of the Cartan lemma
 A: Here's an easier strategy:
Consider the real-analytic vector bundle $\operatorname{Hom}(\mathbb R^n,E)$ where $\mathbb R^n$ denotes the trivial bundle of rank $n=\operatorname{rk}E$.  By assumption, it has a smooth section which is a fiberwise isomorphism.  Approximate this smooth section by a real-analytic section (this is where we use a hard theorem to tell us such an approximation exists).  Now the property of being a fiberwise isomorphism is an open condition, so the real-analytic section is also a fiberwise isomorphism, and thus gives the desired real-analytic trivialization.
A: Here's just a sketch, maybe someone else can fill in the details.  Most of the steps will require the use of the nontrivial fact that $C^\omega(M)$ is dense in $C^\infty(M)$.  Let's assume $M$ is compact.
Step 1: Construct a real-analytic embedding of vector bundles $E\to M\times\mathbb R^N$ for some finite $N$.  This gives a real-analytic classifying map $M\to\operatorname{Gr}_n(\mathbb R^N)$.
Step 2: Increase $N$ to $N'$ so that the composition $\gamma:M\to\operatorname{Gr}_n(\mathbb R^N)\to\operatorname{Gr}_n(\mathbb R^{N'})$ is smoothly null-homotopic.
Step 3: Approximate this smooth null-homotopy of $\gamma$ with a real-analytic homotopy from $\gamma$ to a map $\gamma':M\to\operatorname{Gr}_n(\mathbb R^{N'})$ which is contained in an $\epsilon$-ball over which the universal bundle is real-analytically trivial.
I suspect this argument also works for $M$ non-compact (taking $N=\infty$) though I haven't checked carefully.
