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AlexKontorovich merged 1 commit into from
Dec 19, 2025
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Update L11Lecture.lean #78

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2 changes: 1 addition & 1 deletion Game/Levels/L11Lecture.lean
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Expand Up @@ -28,7 +28,7 @@ Ah, but maybe there's always *some* subsequence that converges? Hmm, but that ca

Oh! But wait, that sequence isn't bounded. Are you saying that if all I know about a sequence is that it's bounded, then there's always *some* subsequence that converges?

**SOCRATES:** Yes, precisely! This important fact is called the \"Bolzano-Weierstrauss theorem\". But here's where it gets **really** subtle. Think about the sequence of fractions: `a (0) = 1 / 1`, `a (1) = 14 / 10`, `a (2) = 141 / 100`, `a (3) = 1414 / 1000`, ... getting closer and closer to $1.4142\\dots = \\sqrt 2$. The sequence is bounded (by $2$, to be crude), and even increasing, but its limit is not a rational number! So the Bolzno-Weierstrauss theorem is not true for the rationals. As I warned you long ago, we'll have to eventually face the fact that we don't even know what the real numbers *are*. I think that time is now.
**SOCRATES:** Yes, precisely! This important fact is called the \"Bolzano-Weierstrass theorem\". But here's where it gets **really** subtle. Think about the sequence of fractions: `a (0) = 1 / 1`, `a (1) = 14 / 10`, `a (2) = 141 / 100`, `a (3) = 1414 / 1000`, ... getting closer and closer to $1.4142\\dots = \\sqrt 2$. The sequence is bounded (by $2$, to be crude), and even increasing, but its limit is not a rational number! So the Bolzano-Weierstrass theorem is not true for the rationals. As I warned you long ago, we'll have to eventually face the fact that we don't even know what the real numbers *are*. I think that time is now.

**SIMPLICIO:** Fine, I'm ready; tell me what they are.

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