I want to prove that $\xi$ is a rank $k$ vector bundle over an $n$ dimensional CW complex $X$ such that $k>n$. Then $\xi$ is trivial iff $\xi \oplus \varepsilon^1$ is trivial. Here $\varepsilon^1$ is the trivial bundle of rank $1$.
Note that one direction is trivial. The non trivial direction is if $\xi \oplus \varepsilon^1$ is trivial implies $\xi$ is trivial.
To prove this I want to use classifying spaces. Let $BO(k)$ be the classifying space of all rank $k$ vector bundles. Now let the bundle $\xi$ be given by the map $$f:X \to BO(k)$$ But we are given that $\xi \oplus \varepsilon^1$ is trivial. This is just pullback of the map $i \circ f$ where $i$ is the map $$i: BO(k) \to BO(k+1)$$ So now since $\xi \oplus \varepsilon^1$ is trivial $i \circ f$ is null homotopic. Now we want to show that $f$ is null homotopic. I am struck here. Any help will be appreciated. Thank you.