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I need to solve 15 Problems for my Real Analysis class all from different topics, problems are from Real Analysis N.L Carothers book, if you have a book I included page #s Given an arbitrary metric space M, show that a decreasing sequence of nonempty compact sets in M has nonempty intersection. (page 113 #31) Let M be a compact and let f;M->M satisfy d(f(x), f(y))>=d(x,y)for all x,y in M. Prove that f is an isometry of M onto itself. (Hint: first, given X in M, consider Xn=f^n(x). By passing to a subsequence, if necessary, we may suppose that (Xn) converges. Argue that Xn->X. Next, given X,Y in M show that we must have d(f(x), f(y))=d(x,y). Thus, f is an isometry onto M. Finally, argue that f has dense range.) (page 114 #42) Given f:R->R and a in R, define F(x)=[f(x)-f(a)](x-a) for x not equal to a. Prove that f is differentiable at a iff F is uniformly continues in some punctured neighborhood of a. (page 116 #52) Show that both sum(from n+1 to infinity)(x^n(1-x)) and sum(n=1 to infinity)(-1)^nx^n(1-x) are convergent on [0,1], but only one converges uniformly. Which one? Why? (page 159 #36) Given a sequence of scalars (Cn) and a sequence of a distinct points (Xn) in (a,b), define f(x)=Cn if X=Xn for some n, and f(x)=0 otherwise. Under what condition(s) is f bounded variation on [a,b]? (page 206 #13) If f,g in Ra[a,b] with f=<g, show that integral a to b(f da)=< integral a to b (g da) (page 218 #1) if f in Ra[a,b], show that |f| in Ra[a,b] and |integral a to b (f da)=<integral a to b |f|da (hint: U(|f|, P)-L(|f|,P)=<U(f,P)-L(f,P) Why?) (page 218 #3) Give an example where f^2 in Ra[a,b] but f is not in Ra[a,b] (page 218 #5) give an example of a sequence of Riemann integrable functions on [0,1] that converges pointless to a nonintegrable function (page 225 #27) If a in Bounded Variation (BV)[a,b], show that Ra[a,b] is a closed subspace of B[a,b]. Specifically, if (fn) is a sequence in Ra[a,b] that converges uniformly to f on [a,b], show that f in Ra[a,b] and that integral a to b (fn da) -> integral a to (f da) (page 231 #40) Given any subset E of R(reals) and any H in R(reals), show that m*(E+h)=m*(E), where E+h={x+h: x in E} (page 271 #4) If E=U(from n+1 to infinity) Xn is a countable union of pairwise disjoint intervals, prove that m*(E)=sum(from n+1 to infinity) L(Xn) (page271 #9) If m*(E)=0, show that m*(E U A)=M*(A)=m*(A | E) for any A (page 271 #16) If E in [a,b] and m*(E)=0, show that E^c is dense in [a,b] (page 271 #17) Find a sequence of measurable sets (En) that decrease to non-empty set, but with m(En)=infinity for all n (page 286 #61) Attached is the same document saved in pages, if you can't open pages I copied and pasted questions above, let me know if you have any questions Thanks