DoWhile

Trigger warning: There's going to be some cryptocurrency-related words in this post. I admit, this is not the best subreddit for this post. But this subreddit has the right kind of people who I want an answer from. Delete this post if it's too out of scope.

Story: I'm happily content with the world of theory surrounding interactive proofs and cryptographic cousins of them. Some that stand out are in the title of this post: VC, WI, and ZK. VC (no privacy, just verify), and ZK (zk simulation privacy) have enjoyed their share of popularity with the cryptocurrency world, whereas WI (verifier can't tell which witness you used in an NP relation proof) holds a place in my theory heart for being

Show that under certain assumptions, every rational number can be represented as x/999...99 for some integer x, and some appropriate number of 9s.

Fun part: What's so special about this representation? In particular, under what assumptions?