Show that A set E subset or equal to R is connected if and only if, for all nonempty disjoint sets A and B satisfying E=A U B there always exists a convergent sequence (x_n)-->x with (x_n) contained in one of A or B and x an element of the other.

29.18
Let f be a differentiable on R with a = sup {|f ′(x)|: x in R} < 1.
Select s0 in R and define sn = f (sn-1) for n ≥ 1. Thus s1 = f (s0), s2 = f(s1), etc
Prove that (sn) is a convergence sequence. Hint: To show (sn) is Cauchy, first show that |sn+1 - sn| ≤ aּ|sn - sn-1| for n ≥ 1.

Please help with the following problem.
Let A be a proper subset of R^m. A is compact, x in A, (x_n) sequence in A, every convergent subsequence of (x_n) converges to x.
(a) Prove the sequence (x_n) converges.
Is this because all the subsequences converge to the same limit?
(b) If A is not compact, show that (a

Prove that a set A, a subset of the real numbers, is compact if and only if every sequence {an} where an is in A for all n, has a convergent subsequence converging to a point in A.
For the forward direction, I know that a compact set is closed and bounded, thus every sequence in A is bounded, and so has a convergent subsequen

(a) Prove this operation:
Let {xn} and {yn} be convergent sequences.
The sequence{zn} where zn:=xn-yn converges and lim (xn-yn)=lim zn=lim xn-limyn
What I attempted was this:
Suppose {xn} and {yn} are convergent sequences and write zn:=xn-yn. Let x:=lim xn, y:=lim yn and z:=x-y
Let epsilon>0 be given. Find M1 s.t. for

See attached file for all symbols.
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? For each of the follwing statements decide if it is true or false. Justify your answer by proving, or finding a couter-example.
1) every bounded sequence of real numbers is convergent.
2) Every convergentsequence is monotone.
3) Every monotone and bounded sequence of real number

Theorem: Suppose that a sequence S of real numbers has a subsequence that converges to a real number a. Then S converges to a.
I know this is true as an if and only if statement, but I need a counter example to show that just one converging subsequence isn't enough.
Here are two sequences I'm considering: {1,-1,1,-1,1,-1..

FUNDAMENTAL MATHEMATICS II
Question 1. Say that a sequence (an) is a Cauchy sequence (named after
the French mathematician Cauchy) if it has the following property:
For every > 0 there is a number M (depending on ) such that
|an - |am < for all n, m >= M.
(1) * Show that the sequences ( 1/n) and (n + 1/2n) and Cauchy s