A. Let f: R--->R and let c be element in R. Show that the lim from x to c of f(x)=L if and only if lim from x to 0 of f(x+c)=L
(if and only if: go both ways)
b. Use either the epsilon-delta definition (which states: Let A be a subset of the reals and let c be a cluster point of A. For a function f: A--->R, a real number L is

The sequence Sn = ((1+ (1/n))^n converges, and its limit can be used to define e.
a) For a fixed integer n>0, let f(x) = (n+1)xn - nxn+1 . For x >1, show f is decreasing and that f(x) . Hence, for x >1;
Xn(n+1-nx) < 1
b) Substitute the following x-value into the inequality from part (a)

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 convergent sequence is monotone.
3) Every monotone and bounded sequence of real number

Please show all work. Please see the attachment for the full problems.
Problem 1 :
Give an example of a sequence {an} satisfying all of the following:
{an} is monotonic
0 < an < 1 for all n and no two terms are equal
=
Problem 2:
Let k > 0 be a constant and consider the important sequence {kn}. It?s behaviou

Let {x_n} be a sequence of positive numbers and suppose that he sequence {x_n+1/x_n} converges to L.
Suppose L <1. Prove that the sequence {x_n} converges to 0.

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.

Attached is a file I need to use to answer some questions. The DNA is 5' to 3'
These questions refer to an intron that starts at nucleotide 128 and is 179 nucleotides long, ending at nucleotide 306.
1. By how many nucleotides do the first six nucleotides (on the 5' end) of the first intron differ from their consensus seque