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 $f_{1}$.