I am reading Bott and Tu Differential Forms in Algebraic Topology. And I want to prove the classification theorem for isomorphism classes of real vector bundles over $M$, i.e:
Theorem 23.12. Let $M$ be a manifold of dimension $m$. There is a one-to-one correspondence $$[M,G_k(\mathbb{R}^{k+m})]\cong Vect_{k}(M)$$
Since two results stated for complex vector bundles (Proposition 23.9 and Lemma 23.9.1) also work for real vector bundles (or at least I think they hold), then I only need to proof the following:
Theorem (I want to prove). Given a manifold $M$ of dimension $m$ and a real vector bundle $E \to M$ of rank $k$. If $n\geq k+m$ and $f,g \colon M \to G_k(\mathbb{R^n})$ are two classifying maps for $E$, then the maps are homotopic.
In Bott and Tu it is given a reference to Stenrood's book but I had a look at it and I would like to avoid that approach.
I would love be given a proof of that, some hints, or (preferably and) a reference where I can read it.
What I have tried: Since the complex case is not proved neither, I have googled for long and I had found one weaker theorem (with two slightly different proofs) which is stated as follows:
Theorem (Weaker). Given a manifold $M$ of dimension $m$ and a real vector bundle $E \to M$ of rank $k$. If $n> k+m$ and $f,g \colon M \to G_k(\mathbb{R^n})$ are two classifying maps for $E$, then the maps are homotopic.
Note the strict inequality n> k+m.
Note: I haven't been able to adapt none of the proofs of the weaker result to the one I want to prove.
I have found that result in this lecture notes and in Hirsch Differential Topology.
Since I have just started studying vector bundles I would like an easy (as rudimentary as possible) approach (so I can fully understand it). Nevertheless, any help would be appreciated.