21. Suppose (An) is a sequence of events. (a) If P (An) → 1 as n → ∞, prove there exists a subsequence {n} tend- ing to infinity such that P(nk Ank) > 0. (Hint: Use Borel-Cantelli.) (b) Show the following is false: Given > 0 such that P(An) ≥ , it follows that there exists a subsequence {n} tending to infinity such that P(nk Ank) > 0.

Q: Let fn RR be a sequence of continuous functions which converg uniformly to a function f : R → R. Let…

A: see my solution

Q: Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is…

A:

Q: n=1 (ii) Give an example such that a, converges conditionally and diverges. IM:

A: Please see the below picture for detailed solution.

Q: 12.4 Show lim sup(sn +tn) N} N}+sup{tn :n > N}.

A:

Q: For each sequence X1, X2, ... of i.i.d. random variables, determine limn→∞ Xn.(a)  Xi ∼Unif(0,…

A: Approach to solving the question:Let's analyze the behavior of each sequence of i.i.d. random…

Q: Is {3n/(3n-1)} n=1 to infinity, a convergent sequence in R with the usual metric dR(x,y) = abs(x-y)?

A: The given sequence is .We need to check that the given sequence is convergent or divergent with the…

Q: x[n]=(0,7)n.u[n] and h[n]=(0,9)n.u[n] to be y[n]=x[n]*h[n] Get the result of my processing…

A: Solution is in step 2

Q: 2. Let fn : [1, 2] → R be a sequence of functions defined by fn(x) = x*. Find the pointwise limit of…

A:

Q: ) Prove that if XnXnand YnYn are any two sequences of real numbers such that XnXn converges to xx…

A:

Q: Suppose real sequences (sn) and (tn) are bounded. (That is, that their ranges are bounded sets.) i.…

A: Bounded sequence and sum of bounded sequences.

Q: Let C be a fixed positive constant. If {bn} is a sequence of nonnegative numbers such that lim (as…

A:

Q: A.) If {a,} is a convergent sequence, then {a,} is also a convergent sequence. B.) If f(x) is a…

A: This question is related to sequence and series.

Q: If an> 0 and Σ an converges, then Σ(-1)"an converges.

A: We have to check wheather it is true or not

Q: 13. Let (an)=I be a Cauchy sequence, and let p : N → N be a one-to-one function. Show that the…

A: A sequence is called a Cauchy sequence if as the sequence progresses, the terms of the sequence…

Q: Examine the increasing and decreasing of each of the following sequences, whether they are bounded…

A:

Q: Let (un) and (Vn) be two sequences defined by n + "n Un+1 Un + Vn 0 < v1 < U1, Un+1 Un + Vn 2 (a)…

A: Given un and vn are two sequences defined by 0&lt;v1&lt;u1, un+1=un2+vn2un+vn, vn+1=un+vn2.

Q: For each of (a)-(d) give a sequence (rn)1 with the stated prop- erties. In each case, the sequence…

A: As per our guidelines we are suppose to answer only 3 sub-parts.

Q: Let (xn) and (yn) be bounded sequences. Show that there exists a common subsequence that each…

A: Sol:- Since both (xn) and (yn) are bounded, we know that there exist M, N &gt; 0 such that |xn| ≤ M…

Q: If lim(x,) =5100, then (Xn) is bounded. Select one: O True O False

A: 1) Here, lim (xn)=5100 that is finite. That is the sequence xn is bounded. So, the statement is…

Q: 10. Let {an}, {bn} and {en} be sequences of real numbers Which of the following statements are true?…

A: To State: True or False. (I) If {an} is convergent and {bn} is bounded then, {anbn} is convergent.…

Q: 1.4 Consider the set S which includes the sequence {(1,),(;, ) ; (§, 1) ; ...: (0 1)}. Now consider…

A: Solution: (a) The set S is S=1,12,12,23,13,34,...,0,1 Clearly, the set S is the collection of open…