Does unbounded curvature implies unbounded diameter?

Question

Let $(M,g)$ be a Riemannian manifold. Assume that for each $C > 0$ there exists $p\in M$ and $X,Y\in T_pM$ unitary such that $K(X,Y) > C$. Does this implies that the diameter of $M$ is infinite?

I just have an intuition about it, related to the this figure:Gabriel's Horn

Answer 1

You could have a sequence of rapidly shrinking spheres all stitched to the same finite open section of a plane. Here, the manifold is not complete though.