If dim$N\geq 2$dim$M$, are Immersions are dense in $C_S^1(M,N)$?

Question

So as the title says the question is that if dim$N\geq$dim $2M$ will immersions be dense in the strong topology? I believe this will be true , since we have that since $dim N\geq 2dim M$ we can view the set of immersions has $\cap_{i=1}^{\infty}X_i$ where $X_i=\{f\in C_S^r(M,N):f|_{K_i} $is an immersion $\}$, where $K_i$ are a compact cover of $M$, and each $X_i$ is a dense open set, since we have that restriction on the dimensions. Now using the fact that $C_S^r(M, N)$ is a Baire space we will have that our desired set will be dense since it's the intersection of dense open sets. What do you guys think? Is there something wrong with this approach? Thanks in advance.