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

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.

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Convergence of a sequence is investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Show that (X, ||*||) is a Banach space if and only if {x in X: ||x||=1} is complete.
Know that in the first direction, we must show that {x in X: ||x||=1} is closed subset of X.
For the reverse direction, I know I have to take a cauchy sequenceand translate it to the unit circle and then show that if it is convergent ther

Use the Secant method (defined in the book) to show that sequence below converges to (square root Q), where Q > 0, given "good" starting values x_0 and x_1:
x_n+1 = (x_n x_n-1 + Q) / (x_n + x_n-1).
Come up with a similar recursion for calculating Q^(1/3) using the secant method.

(x_n tends to plus infinity if and only if for each M belonging to real numbers there exists an N belonging to natural numbers such that:
n>=N implies x_>M.
x_n is said diverge yo minus infinity if and only if for each M belonging to natural numbers such that:
n>=N implies x_n that each of

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

Consider the real sequence {x_n}_n generated by the iteration scheme
x_n+1 = x_n(2-ax_n), for n = 0, 1, 2, ......
where a>0 and x_0 satisfying 0 < x_0 /=x_n>0 for all n.
b. Prove x_n>/=x_n-1.
c. Conclude that lim n-->infinity