Decreasing child mortality rates and increasing healthcare availability is statistically shown to decreased birth rate (among other improvements), resulting in fixing the issue of overpopulation (over time).
Whether it stops at 8 or 9 billion doesn't really make an impact until 2150 at best. IF we had the resources to keep everything puffing till then. We don't. :)
We sure do. The human race will be just fine resource wise.
And that's just accounting for the existing base of resources and knowledge we have today. What we'll have in just another 50 years will be astounding, and further ensure that we won't have a resource problem. :)
You're not even able to moderate yourself to "could" be astounding, it WILL be astounding, categorically? Is there anything pointing toward such being able to reach fruition? Not really.