There is the question, because when we consider the Gromov-Hausdorff distance, we must fix the metric, so we use the natural metric induced from the embedding $S_n \to R_{n+1}$. Is it possible for us to compute the gromov-hausdorff distance $d_{G-H}(S_n,S_m)$ for two different spheres $S_n$ and $S_m$,$m\neq n$?

For example if we want to calculate $d_{G-H}(S_2,S_3)=\inf_{M,f,g}d_{M}(S_2,S_3)$, where $M$ ranges over all possible metric space and $f:S_2\to M$ and $g:S_3\to M$ range over all possible isometric (distance-preserving) embeddings.

At least we can embed $S_2$,$S_3$ into $\mathbb{R}^3$ in a canonical way. This will lead to a upper bound: $g_{G-H}(S_2,S_3)\leq \sqrt{3}$. And in general case we have $d_{G-H}(S_m,S_n)\leq \sqrt{\max(m,n)}$. But it is difficult to get a lower bound control for me. Because we need to take the inf in all possible metric space $M$. Especially I conjecture $d_{G-H}(S_m,S_n)\geq \lambda_{m,n}\sqrt{\max(m,n)}$, where $\liminf_{m,n\to \infty}\lambda_{m,n}>0$.

I only know the knowledge of Gromov-Hausdorff from Peterson's Riemann  Geometry.Unfortunately there is not enough information to compute the gromov-hausdorff distance,so this problem may be very stupid,I will be very appreciate to the pointer.