This is problem 5-4 from Lee's Introduction to Topological Manifolds.

If $M$ is a connected topological manifold and $(p\_1, \dots, p\_k)$ and $(q\_1, \dots, q\_k)$ are two ordered $k$-tuples of distinct points in $M$, then there is a homeomorphism $F: M \to M$ such that $F(p\_i)=q\_i$ for $i=1,\dots ,k$. 

I know how to prove this in the case of $k=1$. By induction, it suffices to prove this in the case $k=2$. I think I have to use the gluing lemma to extend two homeomorphisms each taking $p\_1$ to $q\_1$ and $p\_2$ to $q\_2$, but I am lost here. I would greatly appreciate any help.