Real Analysis UHH
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| A number set is compact iff | it is closed and bounded.
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| [0,1] is | compact
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| C[0,1] is | compact and complete
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| Uniform continuity is | compact
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| Reals are | complete
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| Rationals are not | compact and complete
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| 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|
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| Iff f is continuous | then if U is an open set, then f-1(U) is open relative to D.
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| 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).
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| Iff U is a closed set in f(D) | then f-1(U) is closed relative to D.
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| 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.
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| 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.
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| 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.
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| If sequences are complete | then converge
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| If sequence is Cauchy | then it has no holes
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| 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)]
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| Diatic Rationals (1/2^n) | are dense
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| The statement that a set is dense means | every point in the set is a limit point
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| The statement that a set is dense means | its closure is the whole set
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| Rationals are | dense on the reals
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| Hypothesis for Intermediate Value Theorem | Horizontal line test
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