Disjoint curves in a smooth manifold. I already asked the question here but it was in a too specific way, so I asked it again here for more visibility. How do we prove the following statement :

Let $M$ be a smooth connected manifold without boundary of dim $\geq 2$ and $(x_1, y_1,... , x_n,y_n )$ be $2n$ distinct points of $M$. Then there exist  smooth curves $\gamma_i :[0,1] \rightarrow M$ such that $\gamma_i (0) =x_i$ and $\gamma_i(1) =y_i$ for all $i=1,...,n$ which don't intersect each other.


I would recommend try proving this by induction. Proving that a connect manifold is path connected by a smooth path would be a very useful lemma.
