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Chapter 09Sequences

0109.01Sequence Notation
0209.02Arithmetic and Geometric Sequences
0309.03Recursive Sequences
0409.04The Fibonacci Sequence
0509.05Convergence/Divergence
0609.06Increasing/Decreasing
0709.07Bounded/Unbounded
0809.08The Monotone Convergence Theorem
0909.09Subsequences
1009.10The Bolzano–Weierstrass Theorem
1109.11Cauchy Sequences
1209.12The Cauchy Convergence Criterion

Chapter 09 · Sequences

0109.01Sequence Notation
0209.02Arithmetic and Geometric Sequences
0309.03Recursive Sequences
0409.04The Fibonacci Sequence
0509.05Convergence/Divergence
0609.06Increasing/Decreasing
0709.07Bounded/Unbounded
0809.08The Monotone Convergence Theorem
0909.09Subsequences
1009.10The Bolzano–Weierstrass Theorem
1109.11Cauchy Sequences
1209.12The Cauchy Convergence Criterion

Topics / 09 Sequences

09.10

The Bolzano–Weierstrass Theorem

Every bounded sequence, however wild, has a convergent subsequence.

About 15 minutes

You'll learn to

→State the Bolzano–Weierstrass theorem
→Follow the bisection proof
→Use peaks to find a monotonic subsequence
→Recognize which sequences the theorem applies to

← 09.09 Subsequences 09.11 Cauchy Sequences →

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