0
Add your answer:
Earn +20 pts
Prove that every convergent sequence is a Cauchy sequence?
The limits on an as n goes to infinity is aThen for some epsilon greater than 0, chose N such that for n>Nwe have |an-a| < epsilon.Now if m and n are > N we have |an-am|=|(am -a)-(an -a)|< or= |am -an | which is < or equal to 2 epsilor so the sequence is Cauchy.
What is Example of bounded sequence which is not Cauchy sequence?
((-1)^n)
What Show that 1/2n is a cauchy sequence?
0.5
Examples of bounded and not convergent sequence?
(0,1,0,1,...)
Is it true when a sequence is divergent then its subsequences are divergent explain?
Not always true. Eg the divergent series 1,0,2,0,3,0,4,... has both convergent and divergent sub-sequences.
Any convergent sequence is a Cauchy sequence is converse true?
no converse is not true
Prove that every convergent sequence is a Cauchy sequence?
The limits on an as n goes to infinity is aThen for some epsilon greater than 0, chose N such that for n>Nwe have |an-a| < epsilon.Now if m and n are > N we have |an-am|=|(am -a)-(an -a)|< or= |am -an | which is < or equal to 2 epsilor so the sequence is Cauchy.
What does cauchy constant tells us?
The Cauchy constant, also known as the Cauchy sequence property, tells us that a sequence is convergent if it is a Cauchy sequence. This means that for any arbitrarily small positive number ε, there exists an index after which all elements of the sequence are within ε distance of each other. It is a key property in the study of convergence in mathematics.
What is Example of bounded sequence which is not Cauchy sequence?
((-1)^n)
Is compact metric space is complete?
A compact metric space is not necessarily complete. Compactness only guarantees that every sequence in the space has a convergent subsequence, while completeness requires that every Cauchy sequence converges to a point in the space.
Why set of rational numbers is not complete?
Consider the sequence (a_i) where a_i is pi rounded to the i_th decimal place. This sequence clearly contains only rational numbers since every number in it has a finite decimal expansion. Furthermore this sequence is Cauchy since a_i and a_j can differ at most by 10^(-min(i,j)) or something which can be made arbitrarily small by choosing a lower bound for i and j. Now note that this sequence converges to pi in the reals, so it can not converge in the set of rational numbers. Therefore the rational numbers allow a non-convergent Cauchy sequence and are thus by definition not complete.
What Show that 1/2n is a cauchy sequence?
0.5
What is a cauchy sequence?
(xn) is Cauchy when abs(xn-xm) tends to 0 as m,n tend to infinity.
How do you prove that the sum of a convergent sequence divided by n will converge?
You can use the comparison test. Since the convergent sequence divided by n is less that the convergent sequence, it must converge.
Show that any subsequence of a Cauchy sequence, is couchy?
i don’t know I am Englis
A convergent sequence has a LUB or a what?
JUB
Examples of bounded and not convergent sequence?
(0,1,0,1,...)
