Immersions in Euclidean space are isotopic in its square

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Statement

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

Proof

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