Describe the Dedekind cut corresponding to 2. Show that 2 + 2 = 4 on the level of Dedekind cuts. Explain why the intersection in a metric space M of a closed subset with a sequentially compact subset is sequentially compact. Give an example of a sequence an so that lim sup an does not agree with limn→∞ an, or explain why no such sequence exists. Give an example of an increasing function f : R → R (everywhere de- fined) that fails to be continuous at at least 3 points.

Q: Let a, be recursively defined by ao = 3, an+1 = a - an for n 2 0. Find lim an.

A:

Q: If || fn-fl|p. os no prove that lim 11 fn V = || f 11 p. له در

A:

Q: Recall that Q denotes the set of rational numbers. Show that there does not exist a strictly…

A:

Q: Show that lim cticaN-cLim fu

A:

Q: Evaluate a* -b* lim - b* x→0 X

A:

Q: 8 lim x-1 x-1 (76 - 9x) such that x ER lim(x² – 1) such that x € R x-3

A:

Q: 4. Let an be recursively defined by ao = 3, an+1 = a - = 3, an+1 = a - an for n ≥ 0. Find lim an.…

A: Let  be recursively defined by ,  for .Find .

Q: Let a be any positive real number. Then * 1 lim In a – 1 True False

A:

Q: (4+h)2-16 lim h→0 Compute h

A:

Q: Evaluate the limit V10+102 lim HI00 11+10r

A:

Q: * 15sin(t*)dt Evaluate the limit lim

A:

Q: is Q:- show that the space x = [0₂e) complete space, where e= exponential nat je c= 2.71898

A:

Q: Prove that lim (2a2 + 3r + 6) = 5 using the e, 8 definition of a limit. %3D a -1

A:

Q: 2x² - 1 (a) lim X18 1-x²

A:

Q: lim t→∞ 14/3 + 71/3 2 (6+²/3 +11) ²

A: To find out the limit.

Q: Find the limit HI6) im [In(2 t+x) - In(1+x)] www

A: We will find the limit.

Q: Let A = {(−1)^n + 1/n where n is a positive integer}. Determine A′.(Recall that A′ is the set of…

A:

Q: 6 Use the e - d definition of a limit to prove e lim 1+2 r2 + 2 1.

A:

Q: Suppose that {a} and {b} are two sequences of real numbers where n = 1 to ∞, L and b L, where LER…

A:

Q: Evaluate sin 0 – 1 lim 0 - T/2 In sin 0

A:

Q: Solve using method of undetermined coefficients: y" + y' - 2y = x². %3D

A: Solution :-

Q: Let (an) be a sequence of real numbers that diverges to +∞, and let (bn) be bounded. Prove that lim…

A:

Q: 7. Let A and B be partially ordered classes and let f: A B be a strictly increasing function. Prove…

A: Since, the function f:A→B is strictly increasing for posets A,RA and B,RB, so for any a1,a2∈A and…

Q: Prove that the length of the graph of the Cantor function equals to 2.

A:

Q: Let S be a bounded nonempty subset of R and suppose that inf S /∈ S. Prove that there is a…

A: Step 1: Let s=supS.  So,  for any ε>0, there exists an x∈S such that s−ε<x.  Therefore, for…

Q: lim x18 2x6 - 3x +3 4x5 + 3x4 1

A:

Q: 38

A:

Q: Use the La`Hospital rule to evaluate

A: We have to evaluate the following limit: limt→0et-1-tt2 The above function is in 0/0 form. Therefore…

Q: (a) g: Zg → Z6 defined by g(n) = n 2. (b) g: Z6 → Z6 defined by g(x) =x €x. (c) g: Zs → Zs defined…

A: