Il teorema di Bolzano-Weierstrass

Esistono molte versioni equivalenti del teorema di Bolzano-Weierstrass.
Il teorema fu originalmente provato da Bernard Bolzano nel 1817 e riscoperto da Karl Weierstrass cinquanta anni dopo.

Teorema. Ogni insieme limitato ed infinito contenuto in R ammette punto di accumulazione.

 The greatest lower bound property is equivalent to what is often referred to as the Bolzano–Weierstrass theorem in a form such as ‘a bounded sequence always contains convergent subsequences’. Many introductory textbooks on analysis ask students to accept some further (equivalent) result as an axiom about the real numbers, for example,‘An increasing sequence, bounded above, must be convergent.’