Let $M^{m}$ and $N^{n}$ be $C^{\infty}$ compact manifold with boundary (eventually $\emptyset$) and let $j:M \to N$ be an embedding such that the normal bundle of the embedding $\nu(j)$ is trivial. Let $i$:$M \to \mathbb{R}^{r+m}$ be an other embedding with a sufficiently large $r$. How can I extend $i$ to an embedding from $N \to \mathbb{R}^{r+n}$?
For what I understood a step could be finding an extension of the embedding an open neighborhood of $M$ in $N$ $U \to \mathbb{R}^{n-m} \times \mathbb{R}^{m+r}$ and then extending it to $N$ but I am also having trouble proving this step.