Skip to content

Subsequences

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Keeping infinitely many terms of a sequence, in order. Bolzano–Weierstrass: every bounded sequence of real numbers has a convergent subsequence — the key fact behind the existence of maxima and minima.

Formulas

(a_{n_k})_{k\ge1}, \quad n_1 < n_2 < \cdots

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

Weierstrass extreme value theorem→Rolle's theorem→Mean value theorem→Taylor's theorem and the remainder→Scientific computing★★★★★
Weierstrass extreme value theorem→Rolle's theorem→Mean value theorem→Fundamental theorem of calculus→Symbolic computation (CAS)★★★★★
Weierstrass extreme value theorem→Rolle's theorem→Mean value theorem→Taylor's theorem and the remainder→Floating point (IEEE 754)★★★★★

ℒ AI and machine learning

Weierstrass extreme value theorem→Rolle's theorem→Mean value theorem→Lipschitz continuity→Gradient descent★★★★★
Weierstrass extreme value theorem→Rolle's theorem→Mean value theorem→Fixed-point iteration→Reinforcement learning★★★★★
Weierstrass extreme value theorem→Rolle's theorem→Mean value theorem→Lipschitz continuity→Generative models★★★★★

⚛ Physics and simulation

Weierstrass extreme value theorem→Rolle's theorem→Mean value theorem→Fundamental theorem of calculus→Physics engines★★★★★

⇄ Optimization and systems

Weierstrass extreme value theorem→Operations research and logistics★★★★★

What depends on it

Weierstrass extreme value theorem

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.