2. Let (sn) be a sequence. (a) Prove that limn→oo8n = 0 if and only if...
6. Let si = 4 and sn +1 (sn +-) for n > 0. Prove lim n→oo sn exists and find limn-oo Sn. (Hint: First use induction to show sn 2 2 and the.show (sn) is decreasing)
(2) Let (X, 2, ) be a measure space and let (.). be a sequence of (E, BR)-measurable functions from X to R. Suppose that 'n pointwise and lim. S du SI du < 0. Prove that for all E E lim,- Jef du = sef du < 0.
2. (5 points) Let {sn}nen be a sequence. Let S be the set of subsequential limits of {Sn}nen, that is, x E S if and only if 3{Sn}ken subsequence of {Sn}nen such that limky Sny = x. Use the previous problem to show that inf S = lim inf sns sup S = lim sup sn.
Please answer all parts. (2) (a) Give an example of sequences (sn) and (tn) such that lim sn ntoo 0, but the sequence (sntn) does not converge does not converge.) (b) Let (sn) and (tn) be sequences such that lim sn (Prove that it O and (tn пH00 is a bounded sequence. Show that (sntn) must converge to 0. 1 increasing subsequence of it (b) Find a decreasing subsequence of it (3) Consider the sequence an COS (а) Find an...
(6) Let (, A,i) be a measure space. Let fn : 0 -» R* be a sequence of measurable functions. Let g, h : O -> R* be a pair of measurable functions such that both are integrable on a set A E A and g(x) < fn(x)<h(x), for all E A and ne N. Prove that / lim sup fn du fn dulim sup fn du lim inf fn du lim inf n o0 A n-oo A noo n00...
Let (an) be a sequence such that lim an = 0. Define the sequence (AR) Exercise 21: by A =ļa, and An = zou-a + ax=a + zam for k21. Prove that an converges to some S if and only if Ax converges to S. N=0 k=0 Exercise 22: (Cauchy condensation test) Let (an) be a sequence such that 0 < antı san a) Show n=0 n=1 Hint: Recall the proof of convergence of for p > 1. Ren for...
#s 2, 3, 6 2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
3) Let (an)2- be a sequence of real numbers such that lim inf lanl 0. Prove that there exists a subsequence (mi)2-1 such that Σ . an, converges に1
Let (, A, ) be a measure space. Let fn : 2 -» R* be a sequence of measurable functions. Let g,h : 2 -» R* be a pair of measurable functions such that both are integrable on that a set A E A and g(r) fn(x)h(x), for all E A and nE N. Prove / fn du lim sup A lim inf fn dulim inf lim sup fn du A fn du no0 no0 A noo n+o0 (You may...
IDY in < oo and lim - Yn < 0o. Prove that lim,+ 1. Let In > 0. Yn > 0 such that lim,- Yn) < lim,-- In lim,+ Yn: i tn < oo and lim yn < . Prove that lim. In 1. Let In 20, yn 0 such that lim Yn) < limn+In lim + Yr