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Question

Let $\pi:M\to N$ be a surjective smooth submersion between manifolds. If $y_n\in N$ is a convergent sequence, is there a convergent sequence $x_n\in M$ such that $\pi(x_n)=y_n$?

Attempt at answering my question

Convergence is a local property. Moreover $\pi$ is locally a projection $$R^r\times R^s\to R^r.$$ Thus, we can just take $x_n=(y_n,0)$. Done.

Right?

A generalization of that question

Was the assumption that $\pi$ is a surjective smooth submersion really necessary or does that hold under weaker hypothesis?

Raoul. P. Q.
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    You need to be a bit more careful; your definition only makes sense in a neighborhood of the limit $\lim_{n \to \infty} y_n = y \in N$ where $\pi$ is in fact locally a projection, but $y_n$ may start out outside this neighborhood. – Qiaochu Yuan Apr 30 '16 at 16:04

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