Existence of sections of sphere bundles. I am reading this article, could you please tell me where I can find the proof of Theorem 2.1.1?

Theorem 2.1.1. Let $\pi: S \rightarrow M$ be a $(d-1)$-sphere bundle over a closed manifold $M$ of dimension $d$. Suppose that the structure group of $S$ can
be reduced to the orthogonal group $\mathrm{O}(d, \mathbb{R})$, then there exists a smooth map $s: M \rightarrow S$ such that $s \in \Gamma(M \backslash I, S)$, where $I$ is discrete.

Thanks!
 A: Here is the correct statement:
Theorem. Suppose that $M$ is a smooth closed $n$-manifold, $\xi=(\pi:S\to M)$ is a smooth fiber bundle with fibers $S^{n-1}$, whose structure group is reduced to the orthogonal group $O(n)$. Then there is a finite subset $I\subset M$ and a smooth section $s$ of this bundle over $M\setminus I$.
Proof. By the assumption, the bundle $\xi$ is associated with a principal $O(n)$-bundle over $M$, hence, there is a rank $n$ real vector bundle $\eta=(E\to M)$ equipped with a (fiberwise) metric, such that $\xi$ is the unit sphere bundle of $\eta$.
Now, we can start the actual proof. First, take the zero section $\sigma_0$ of $\eta$. Then perturb it to a smooth section $\sigma$ of the same bundle $\eta$ which is transversal to the zero section. (Such perturbation arguments you can find in any good differential topology textbook, say, in Guillemin and Pollack. Incidentally, it is a good exercise to see that if $\sigma$ is a smooth map $M\to E$ sufficiently close to a section in $C^{1}$ topology, then $\sigma$ itself is the precomposition of a section with a diffeomorphism $M\to M$ close to $id_M$.) By transversality and since $E$ has dimension $2n$, it follows that the set of zeroes of $\sigma$ is a 0-dimensional submanifold $I$ in $M$. By compactness of $M$, $I$ has to be finite. Now, on $M\setminus I$, replace $\sigma$ with
$$
s=\sigma/|\sigma|,
$$
where the norm is the fiberwise norm on $\eta$. Thus, $s$ is the required section of $\xi$. qed.
The argument I gave is very standard, it is used for instance to define index of a vector field, in the case when $\eta$ is the tangent bundle of $M$.
