Question

Let (xn) be Cauchy in (M, d) and a ∈ M. Show that the
sequence

d(x_{n}, a) converges in R. (Note: It is not given that
x_{n} converges to a.

Hint: Use Reverse triangle inequality.)

Answer #1

Exercise 2.4.5: Suppose that a Cauchy sequence {xn} is such that
for every M ∈ N, there exists a k ≥ M and an n ≥ M such that xk
< 0 and xn > 0. Using simply the definition of a Cauchy
sequence and of a convergent sequence, show that the sequence
converges to 0.

Let X = (xn) be a sequence in R^p which is convergent
to x. Show that lim(||xn||) = ||x||. hint: use triange
inequality

Use the definition of a Cauchy sequence to prove that the
sequence defined by xn = (3/2)^n is a Cauchy sequence in R.

Let
<Xn> be a cauchy sequence of real numbers. Prove that
<Xn> has a limit.

If Xn is a cauchy sequence and Yn is also a cauchy sequence,
then prove that Xn+Yn is also a cauchy sequence

Let 0 < θ < 1 and let (xn) be a sequence where
|xn+1 − xn| ≤ θn for n
= 1, 2, . . ..
a) Show that for any 1 ≤ n < m one has |xm −
xn| ≤ (θn/ 1-θ )*(1 − θ m−n ).
Conclude that (xn) is Cauchy
b)If lim xn = x* , prove the following error in
approximation (the "error in approximation" is the same as error
estimation in Taylor Theorem) in t:...

if
{Xn} and {Yn} are cauchy, show that {Xn +Yn} is cauchy.
b.) Also show that {XnYn} is cauchy

Let f: (0,1) -> R be uniformly continuous and let Xn be in
(0,1) be such that Xn-> 1 as n -> infinity. Prove that the
sequence f(Xn) converges

Let xn be a sequence such that for every m ∈ N, m ≥ 2 the
sequence limn→∞ xmn = L. Prove or provide a counterexample: limn→∞
xn = L.

Let {Xn} be a sequence of random variables that follow a
geometric distribution with parameter λ/n, where n > λ > 0.
Show that as n → ∞, Xn/n converges in distribution to an
exponential distribution with rate λ.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 10 minutes ago

asked 15 minutes ago

asked 17 minutes ago

asked 18 minutes ago

asked 24 minutes ago

asked 26 minutes ago

asked 29 minutes ago

asked 33 minutes ago

asked 38 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago