This, plus hours of Racket coding ahead of me, makes for a great Monday afternoon :)

(1 + 1 / x)^x approaches e as x approaches infinity. So (1 + 1 / lg n)^(lg n) is about e, and n is (lg n) (n / lg n), and the statement follows. You can make this more precise by using inequalities, but there’s also probably some imprecision in your algorithm analysis already. A lot of algorithm analysis is about making O-notation statements that you hope no one will challenge you on, because it’ll take you a week to justify them formally.

(Quick)script of the week: “Open terminal here” https://github.com/Metaxal/quickscript-extra/blob/master/scripts/open-terminal.rkt recently updated to support MacOS and Windows, you can get it by installing or updating the quickscript-extra collection, or just copy the source directly into your DrRacket script library.