8. Show that, at the n stage of the construction of the Cantor Middle- Thirds set, the sum of the lengths of the remaining intervals is Conclude that the sum of the lengths of these intervals tends to 0 as n → ∞o.

Q: 12.Q. Let 0 < aı < bị and define az = Va,b , b2 Vanba , bat1 (a, + bi)/2, ..., Ant1 = (an + bn)/2,…

A:

Q: n 11. Let X be a non-empty set that is bounded from below. Show that there is a decreasing sequence…

A:

Q: Suppose that the sequence x0, x₁, x2... is defined by x = 6, x₁ = 3, and xk+2 = 2xk+1+3xk for k≥0.…

A:

Q: = 5. Prove that if x1 = 1 and xn+1 = V6 + xn, then xn 1. e 6. Let E CR such that E is nonempty and…

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Let (an) be the sequence with an = (-1)” +212/2. (i) Use the fact that 1/n² → 0 as n → ∞ and…

A:

Q: 9. Use Cantor's diagonal argument to show that the set of all infinite sequences of Os and 1s (that…

A: please see the next step for solution

Q: 7. Suppose A and B are compact subsets of R and that fn is a sequence of functions converging…

A:

Q: 1. Give an example of sequence of real numbers with the following properties.  a) Cauchy but not…

A:

Q: Suppose that the sequence x0, x1, x2... is defined by xo = 5, x₁ = 5, and xk+2 = −8xk+1-15xk for…

A:

Q: -5/7-3x with two intervals 36/5 and 19/5

A: For two Intervals  (36/5, 19/5) For X= 36/5 The Value of given expression is,

Q: 3.5.4) Show directly from the definition that of (xn) and (yn) Cauchy sequences, then cxntyn) and…

A:

Q: 3. Prove that if (a, } and (b,} are Cauchy sequences of rational numbers satisfying lim (a, - b, = 0…

A: 3.

Q: 1. Consider the following sequence xn = 여 1 1 4m, if n is odd 4n-2, if n is even Compute lim sup and…

A: Answer lim sup=lim inf=0  as  n  goes to infinity.Explanation:Step 1: Solution : Step 2:

Q: Determine whether an = limit if it converges. In (5n+2) In (2n-3) converges or diverges and…

A:

Q: The set of all rational numbers in the interval (0,1) is countable, therefore it can be arranged…

A:

Q: B Let r and y. be any two sequences of real numbers. Prove: If the surm rn+un converges and if the…

A:

Q: 2. a) Suppose (pn) has limit L, and let S be the set of values of the terms of (pn). Can we say that…

A: a) Suppose pn has a limit say L. That is, pn→L. For given ε&gt;0, there exists n0∈ℕ such that…

Q: Consider the sequence b = 1+tann 11 . In order to apply the squeeze theorem and determine the limit…

A:

Q: 3. Suppose C₁, C2, C3, ... is a non-decreasing sequence of sets, i.e., Ck C Ck+1, Ck is defined as…

A:

Q: (b) From part (a), we know that lim an exists. Compute it. n00

A: Given that a1=3 and an+1=12+an for each n≥1 Also, the sequence an is monotonic increasing and…

Q: 5. Show that the following sequences of sets, {Ck }, are non-decreasing, (1.2.16), then find limg→∞…

A: