I am trying to do the following exercise from Hirsch's Differential topology :
An immersion $f:M\rightarrow N$ is in general position if for ant integers $k\geq 2$, whenever $f(x_1)=...=f(x_k)=y$ and the points $x_1,..,x_k$ are disticnt , then $T_y N = T_{x_k} f + \cap _{i=1}^{k-1}T_{x_i}f$. Then the set of proper immersions which are in general position is open and dense in $Imm_S^r(M,N)$.
Now starting to try and to prove this is open , there will be two cases one if $f$ is injective and the other if it isn't . If $f$ is injective we have that a proper injective immersions is an embedding and so we get an embedding and we can use the fact that these are open . If $f$ is not injective here is where I come intro trouble, I don't know how to find a neighborhood for this function that lies inside the desired set. I have tried using the fact that locally there are neighborhoods where the function looks like an inclusion but I got nowhere, I also tried to see this in terms of transversality but I can't seem to find a place where to use this, also I am kinda lost when it comes to be the density part. Any hints or comments are appreciated. Thnaks in advance.
Idea : Try and use the fact that $Imm^r(M,N)$ is a Baire space, and show that that our desired set is residual, but I haven't been able to do this .
Does anyone have any suggestions?