(a) Prove that the norm limit of a sequence of compact operators is compact.
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- Find and prove the limit or deteremine if the limit doesn't exist: n/(n2n-5)Let (an) be a sequence of real numbers that diverges to +∞, and let (bn) be bounded. Prove that lim (an + bn) = +∞.Consider the space Z+ with the finite complement topology. Consider the sequence (xn) of points in Z+ given by xn = n+7. To what point or points does the sequence converge?
- (a) Prove that a bounded function f is integrable on [a, b] if and only if there exists a sequence of partitions (Pn)∞n=1 satisfyingUse a Karnaugh map to write a Boolean equation for the function in minimzed sum of productsProve that limits of sequences are unique. That is, show that if L1 and L2 are numbers such that an → L1 and an → L2, then L1 = L2.
- Let p ≥ 1 and lp be the set of all sequences x = (x1, x2, · · ·) of real numbers suchthatkxkp =Xi|xi|p1/p< ∞.Show that lp with p 6= 2 is not an inner product space.Let A be a nonempty subset of R that is bounded above and let α = sup A. If α is NOT an element in A, prove that there existsa sequence {xn} in A with xn → α.Prove that a set T1 is denumerable if and only if there is a bijection from T1 onto a denumerableset T2.