# proof of Cauchy condition for limit of function

The forward direction is . Assume that ${lim}_{x\to {x}_{0}}f(x)=L$. Then given $\u03f5$ there is a $\delta $ such that

$$ |

Now for $$ and $$ we have

$$ |

and so

$$ |

We prove the reverse by contradiction^{}.
Assume that the condition holds.
Now suppose that ${lim}_{x\to {x}_{0}}f(x)$ does not exist. This means that for
any $l$
and any $\u03f5$ sufficiently small then for any $\delta >0$ there is
${x}_{l}$ such that $$.
For any such $\u03f5$ choose $u$ such that $$ and
put $l=f(v)$ then substituting in the condition with $u={x}_{l}$ we get
$$. A contradiction.

Title | proof of Cauchy condition for limit of function |
---|---|

Canonical name | ProofOfCauchyConditionForLimitOfFunction |

Date of creation | 2013-03-22 18:59:08 |

Last modified on | 2013-03-22 18:59:08 |

Owner | puff (4175) |

Last modified by | puff (4175) |

Numerical id | 8 |

Author | puff (4175) |

Entry type | Proof |

Classification | msc 54E35 |

Classification | msc 26A06 |

Classification | msc 26B12 |