Theory of complexity – Prove $ mathsf {DSPACE} ( log log log n) = mathsf {DSPACE} (1) $

There is a similar question here
. Instead of proving $ mathsf {DSPACE} ( log log log n) = mathsf {DSPACE} (1) $, the attached link is trying to prove $ mathsf {DSPACE} (o ( log log n)) = mathsf {DSPACE} (O (1)) $. My idea is to solve this question using the same logic. However, I also can not understand the explanation of the evidence in this message, nor relate the proof of this message to my question here.

Could someone give me explanations or advice on solving this problem?

Any help would be appreciated.

Thank you in advance.