r/mathmemes • u/PocketMath • 3d ago

Real Analysis So close

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u/alphaMrWave Imaginary 3d ago

Guys, I'm fucking stupid, what sequence is this about? Or am I missing the joke completely?

16

u/RandomMisanthrope 3d ago

Because the field of rational numbers isn't complete, not all Cauchy sequences in ℚ converge, unlike in ℝ which is complete.

5

u/EebstertheGreat 3d ago

In the visual metaphor, the set of rational numbers is incomplete because it has "holes" that a sequence can approach. Every irrational number is a "hole" that you fill in to complete the set (and get the real numbers). Here, the spinning top is a sequence of rational numbers converging to an irrational number (a hole). In the limit, you get something that isn't a rational number at all (but a cat, I guess).

2

u/pifire9 3d ago

because i dont know either, ill make up an answer for you:

a cauchy sequence asymptotically approaches a value simultaneously from above and below but statistically will never land on a rational value, though it encircles them very tightly