# Every immersion can be deformed to have only transverse self-intersections

I asked this question some time ago in MSE, but obtained no answer. Maybe this is the right place to post it.

Let $$f : M^n \to \overline{M}^{n+k}$$ be an immersion between smooth manifolds. Is it true that there exists a smooth map $$F : M \times [0,1] \to \overline{M}$$ such that the following conditions hold?

1. $$F_0 = f$$;
2. $$F_t : M \to \overline{M}$$ is an immersion for every $$t \in [0,1]$$;
3. $$F_1(M)$$ has only transverse self-intersections.

(Here, $$F_t(p) = F(p,t)$$ for every $$(p,t) \in M \times [0,1]$$.)

If this does not hold in this full generality, is it true for hypersurfaces ($$k=1$$)? For $$\overline{M} = \mathbb{R}^{n+k}$$?