Ask Dr. Analysis: Is This Set Dense?
An advice-column epsilon judges which sets of real numbers are dense, ending in an outburst about a set it calls full of gaps.
Transcript
[Title] "This week on Ask Dr. Analysis: 'Is this set dense in the real numbers?'" [An ε character answers]
Numbers whose decimal expansions contain the digit sequence "999": "Yes, dense." · …contain at least one million consecutive 9's: "Dense, obviously." · …contain at least a million sequences of at least a million consecutive 9's: "C'mon. Still dense. Those numbers are everywhere." · …contain no more than a million sequences of a million consecutive 9's: "NOT DENSE!! NOT DENSE!!! YOU COULD DRIVE A TRUCK THROUGH THOSE GAPS!!!"
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Free for classrooms, worksheets, slides and other non-commercial use under CC BY-NC 4.0, with credit to Ben Orlin.
Cartoon by Ben Orlin, Math with Bad Drawings. https://cartoons.mathwithbaddrawings.com/2024-10-01-ask-dr-analysis-is-this-set-dense/ (CC BY-NC 4.0)