I am trying to prove that there exists a formal immersion from a manifold $M$ of dim $m$ into $\mathbb{R}^{2m}$. Formal immersion is just an injective bundle map from $TM$ into $T\mathbb{R}^{2m}$ that is, injective on the fibres.
For proving this all I have to show is I can find a bundle of rank $m$ such that the sum is trivial. I can prove that there exists a bundle such that sum is trivial but I am unable to ensure that the rank is less than $m$. if anyone can give some hint to solve this it would be great. Thanks.