Theorem: Every Bounded Real Sequence has a Convergent Subsequence (Bolzano, Weierstrass) 

Every bounded real sequence \((a_n)_{n\in\mathbb N}\) has a convergent subsequence \((a_{n_k})_{k\in\mathbb N}\) (i.e. at least one accumulation point).

Notes

• The theorem is named after Bernard Bolzano and Karl-Theodor Weierstrass.
• The theorem can be also formulated for general metric spaces.

| | | | | created: 2015-02-28 13:40:52 | modified: 2017-04-16 11:48:59 | by: bookofproofs | references: [581]

1.Proof: (related to "Every Bounded Real Sequence has a Convergent Subsequence")