More detail please on exactly why the Hausdorff measure is monotone decreasing?

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I've seen similar questions to my question, as has been suggested by some responses to the first version, but the answers (e.g. from Christopher L), don't help me understand exactly why the Hausdorff measure decreases as delta increases. In particular I'm not able to see why the infimum of the sum of the diameters of the cover could become smaller.

Please could someone spell it out for me.

Thank you.