Suppose we're in a general normed space, and we're considering a sequence [itex]\{x_n\}[/itex] which is bounded in norm: [itex]\|x_n\| \leq M[/itex] for some [itex]M > 0[/itex]. Do we know that [itex]\{x_n\}[/itex] has a convergent subsequence? Why or why not?(adsbygoogle = window.adsbygoogle || []).push({});

I know this is true in [itex]\mathbb R^n[/itex], but is it true in an arbitrary normed space? In particular, since it's true in [itex]\mathbb R[/itex], we know that [itex]\{\|x_n\|\}[/itex] has a convergent subsequence [itex]\{\|x_{n_k}\|\}[/itex] that converges to some [itex]z\in \mathbb R[/itex]. My first instinct would be to try to apply the triangle inequality to show that [itex]\|x_{n_k} - x\| \to 0[/itex] for [itex]\|x\| = z[/itex], but the triangle inequality doesn't give me what I want here, since I need to bound [itex]\|x_{n_k} - x\|[/itex] by something that can be made arbitrarily small, but I only have [itex]| \|x_{n_k}\| - \|x\| | \leq \|x_{n_k} - x\|[/itex].

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Bounded sequences and convergent subsequences in metric spaces

Loading...

Similar Threads for Bounded sequences convergent |
---|

I arctan convergence rate |

I Convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## |

A Getting a finite result from a non-converging integral |

I Convergence of a recursively defined sequence |

I What is this sequence that converges to ln(x) called? |

**Physics Forums | Science Articles, Homework Help, Discussion**