# proof of Dini’s theorem

Without loss of generality we will assume that $X$ is compact^{} and, by replacing
${f}_{n}$ with $f-{f}_{n}$, that the net converges^{} monotonically to 0.

Let $\u03f5>0$.
For each $x\in X$, we can choose an ${n}_{x}$, such that $$. Since ${f}_{{n}_{x}}$ is continuous^{},
there is an open
neighbourhood ${U}_{x}$ of $x$, such that for each $y\in {U}_{x}$, we have $$. The open sets ${U}_{x}$ cover $X$, which is compact, so we can choose
finitely many ${x}_{1},\mathrm{\dots},{x}_{k}$ such that the ${U}_{{x}_{i}}$ also cover $X$. Then,
if $N\ge {n}_{{x}_{1}},\mathrm{\dots},{n}_{{x}_{k}}$, we have $$ for each
$n\ge N$ and $x\in X$, since the sequence ${f}_{n}$ is monotonically decreasing.
Thus, $\{{f}_{n}\}$ converges to 0 uniformly on $X$, which was to be proven.

Title | proof of Dini’s theorem |
---|---|

Canonical name | ProofOfDinisTheorem |

Date of creation | 2013-03-22 12:44:13 |

Last modified on | 2013-03-22 12:44:13 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Proof |

Classification | msc 54A20 |