Immersions in Euclidean space are isotopic in its square

Statement

Suppose $f_0,f_1:M\to {\mathbb{R}}^n$ are two immersions. Consider ${\mathbb{R}}^n$ embedded inside ${\mathbb{R}}^{2n}$. Then, the induced immersions of $f_0,f_1$ in ${\mathbb{R}}^{2n}$ are isotopic as immersions in ${\mathbb{R}}^{2n}$.

Proof

The key idea of the proof is to swing $f_0$ from the first factor ${\mathbb{R}}^n$ to the second factor ${\mathbb{R}}^n$, and then swing back, gradually changing $f_0$ to $f_1$.