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Prove the inequality
$d_H(\mathrm{spt(\mu),\mathrm{spt(\nu))\leq W_\infty(\mu,\nu)$

where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\infty$ is the Wasserstein infinity distance between $\mu$ and $\nu$. I believe this is obvious because min max > max min, but this seems to be different. Any guidance on how to tackle this inequality would be greatly appreciated. Please let me know if I need to provide any additional context.

Prove the inequality
$$d_{H}(\mathrm{spt}(\mu),\mathrm{spt}(\nu))\leq W_{\infty}(\mu,\nu)$$

where $d_H$ denotes the Hausdorff distance between the supports of the measures $\mu$ and $\nu$, and $W_\infty$ is the Wasserstein infinity distance between $\mu$ and $\nu$. I believe this is obvious because min max > max min, but this seems to be different. Any guidance on how to tackle this inequality would be greatly appreciated. Please let me know if I need to provide any additional context.