Let $f\colon N^n\to M^{2n}$ be an immersion. Then, we can extend $f$ to $\bar{f}\colon E(\nu_g)\to M$ of the total space of the normal bundle.

Let $s_0\colon N\to E(\nu_g)$ be a zero section and $s_1$ be the section which is zero at only a finite number of points. Define an isotopy $s_t\colon N\to E(\nu_g)$ between $s_0$ and $s_1$.

Let $f_t=\bar{f}\circ s_t\colon N\to M$. Then, $f_0=f$ by our construction.

Suppose that $f_0(x)=f_0(y)$ for $x\neq y$ is it ture that there exists a unique $y_1\in N$ in the neighborhood of $y\in N$ such that $f_0(x)=f_0(y)=f_1(y_1)$?