# proof of Weierstrass M-test

Consider the sequence of partial sums ${s}_{n}={\sum}_{m=1}^{n}{f}_{m}$. Take any $p,q\in \mathbb{N}$ such that $p\le q$,then, for every $x\in X$, we have

$|{s}_{q}(x)-{s}_{p}(x)|$ | $=$ | $\left|{\displaystyle \sum _{m=p+1}^{q}}{f}_{m}(x)\right|$ | ||

$\le $ | $\sum _{m=p+1}^{q}}|{f}_{m}(x)|$ | |||

$\le $ | $\sum _{m=p+1}^{q}}{M}_{m$ |

But since ${\sum}_{n=1}^{\mathrm{\infty}}{M}_{n}$ converges, for any $\u03f5>0$ we can find an $N\in \mathbb{N}$ such that, for any $p,q>N$ and $x\in X$, we have $$. Hence the sequence ${s}_{n}$ converges uniformly to ${\sum}_{n=1}^{\mathrm{\infty}}{f}_{n}$.

Title | proof of Weierstrass M-test |
---|---|

Canonical name | ProofOfWeierstrassMtest |

Date of creation | 2013-03-22 12:58:01 |

Last modified on | 2013-03-22 12:58:01 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 5 |

Author | CWoo (3771) |

Entry type | Proof |

Classification | msc 30A99 |

Related topic | CauchySequence |