What are examples of non-compact complete Riemannian manifolds with everywhere positive curvature?
Can you give examples of 2-dimensional surfaces in $\mathbb{R}^3$ with this property?
Note that by Bonnet-Myers theorem, if the curvature is bounded from below by a positive number, then the manifold is compact, so the curvature must decrease to 0.
Think about a paraboloid like $z=x^2+y^2$. The idea is that the curvature is positive but tends towards zero as we take $x^2+y^2=r^2$ to infinity. This is a good calculation to work out as an example.