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# Real Analysis UHH

Question | Answer |
---|---|

A number set is compact iff | it is closed and bounded. |

[0,1] is | compact |

C[0,1] is | compact and complete |

Uniform continuity is | compact |

Reals are | complete |

Rationals are not | compact and complete |

Iff f is continuous | then if p is in D and c>0 then there is a d>0 such that if x is in D and |x-p| |

Iff f is continuous | then if U is an open set, then f-1(U) is open relative to D. |

Iff f is continuous | then if p is in D and S is a sequence in D with limit p, then the sequence T defined by T(n)=f(S(n)) has limit f(p). |

Iff U is a closed set in f(D) | then f-1(U) is closed relative to D. |

If f is continuous over the compact set A and c > 0 | then there is a d > 0 such that if each of x and y is in A and |x - y| < d, then |f(x) - f(y)| < c. |

Let D be a subset of the reals and for each positive integer i, let fi:D R. The statement that this sequence of functions converges pointwise means that | for each p in D the number sequence Sp defined by Sp(n) = fn(p) has limit. |

The statement that the function f:[a,b]-->R is of bounded variation (denoted by B.V.) means | there is a number M > 0 such that if x1, x2,...,xn is a subdivision of [a,b] then Integral < M. |

If sequences are complete | then converge |

If sequence is Cauchy | then it has no holes |

The statement that the sequence of functions f=f1,f2,.... converges uniformly to f:[a,b]-->R means | if c>0, there exists a N∈Z+ s.t. if i>N and x∈[a,b], then [fi(x)-f(x)] |

Diatic Rationals (1/2^n) | are dense |

The statement that a set is dense means | every point in the set is a limit point |

The statement that a set is dense means | its closure is the whole set |

Rationals are | dense on the reals |

Hypothesis for Intermediate Value Theorem | Horizontal line test |

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