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# Real Analysis UHH

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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