differential geometry – Closed embedding of the half line in a non-compact manifold


I am trying to do an exercise for some time now that is the following :

Let $M$ be a connected Hausdorff non-compact paracompact $C^r$ manifold. Then there is a closed $C^r$ embedding of the half line $[0,infty[$ into $M$.

I have tried two approaches to this

  1. I consider a compact exhaustion of $M$ and try and create this map by considering injective paths between points in $K_i-intK_i$ and such. Problem for this I can’t control the injectivity of the different paths.

2.I consider a chart that has no finite subcover , I can assume this has countable subcover tought since $M$ is secound-countable. Now I try usnig these charts and open sets construct path betwenn the points myself that are injective, but I always run into the same problem that I don’t seem to be able to control the injectivity for the different paths.

The two approaches are kinda similar but I don’t know what else could be done.

Any help with this is aprecciated. Thanks in advance.