I recently asked this question Unbounded sectional curvature implies infinite diameter?. I would like now to ask something similar, but in another context.
Suppose you have a complete metric space $(M,d)$. Assume that the curvature of $M$ nas no upper bound. Can one concludes that the diameter of $M$ is infinite?
If it helps, one can assume that $(M,d)$ is the limit of a sequence of compact manifolds.
Sorry for being vague on the first time. I am assuming I have a length space $(M,d)$ which is actually complete. It is obtained as the Gromov-Hausdorff limit of a sequence of compact Riemannian manifolds. So my question is: if the Alexandrov curvature of $(M,d)$ is not bounded from above, is the diameter of $(M,d)$ infinite?