Why are normal bundles always locally trivial?

Question

Is there a quick and dirty proof that normal bundles (say of some submanifold in a smooth manifold) are always locally trivial?

My notes seem to have swept this assumption under the rug. Even pointing me to a source would be fine.

Answer 1

Edited the answer to deal with Marie's objections in the comments.

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Let $M$ be an $m$-manifold and $i: N \to M$ an immersed $n$-submanifold. Let us prove local triviality of the normal bundle at a point $x \in N$.

By the implicit function theorem, co-ordinate charts $U$ around $x$ and $V$ around $i(x)$ exist so that the embedding looks like the mapping $(x_1, x_2, \ldots , x_n) \mapsto (x_1, x_2, \ldots , x_n, 0, 0, \ldots)$ from $\mathbb R^n$ to $\mathbb R^m$, where the image of the mapping has the last $m-n$ co-ordinates to be $0$.

Look at what is happening at $U$. Let $T_M(U)$ be the pullback of the tangent bundle of $M$ to $U$, and similarly for $T_N(U)$. The normal bundle $B$ on $U$ is the quotient bundle $T(M)/T(N)$. As seen by the nature of embedding which is explicitly given by the vanishing of the last few co-ordinates, this quotient is also a trivial bundle.

So the normal bundle is locally trivial. Note the use of the implicit function theorem.