Question

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

Answer #1

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

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.

Let
( xn) and (yn) be sequence with xn converge to x and yn converge to
y. prove that for dn=((xn-x)^2+(yn-y)^2)^(1/2), dn converge to
0.

Prove that if a sequence is a Cauchy sequence, then it
converges.

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 (xn) be Cauchy in (M, d) and a ∈ M. Show that the
sequence
d(xn, a) converges in R. (Note: It is not given that
xn converges to a.
Hint: Use Reverse triangle inequality.)

(a) Is the converse of Bolzano-Weierstrass Theorem true? If yes
prove it. If false provide a counterexample.
(b) Since Q is countably infinite, it can be written as a sequence
{xn}. Can {xn} be monotone? Briefly
explain. Hint. Assume it’s monotone, what would be the
consequences?
(c) Use the , N definition to prove that if {xn} and {yn} are
Cauchy then {xn + yn} is Cauchy too.

Prove: Let x and y be bounded sequences such that xn
≤ yn for all n ∈ N. Then lim supn→∞
xn ≤ lim supn→∞ yn and lim
infn→∞ xn ≤ lim infn→∞
yn.

Given that xn is a sequence of real numbers. If (xn) is a
convergent sequence prove that (xn) is bounded. That is, show that
there exists C > 0 such that |xn| less than or equal to C for
all n in N.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 4 minutes ago

asked 10 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 24 minutes ago

asked 33 minutes ago

asked 36 minutes ago

asked 45 minutes ago

asked 46 minutes ago

asked 54 minutes ago

asked 1 hour ago

asked 1 hour ago