How do I construct an immersion $\mathbb{R}\to\mathbb{R}^2$ that can't be approximated by an embedding? If the dimensions of the spaces would not fulfill the condition $2\cdot m=2\cdot 1=2=n$ there would be an approximation (cf. Whitney immersion theorem).
I know that there is this post but frankly I don't understand the example that was given and there the immersion was defined on $S^1$ which is a one dimensional manifold but I need to construct one on $\mathbb{R}$.
Does someone know another example?