Question

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.

Answer #1

Define a sequence (xn)n≥1 recursively by x1 = 1, x2 = 2, and xn
= ((xn−1)+(xn−2))/ 2 for n > 2. Prove that limn→∞ xn = x exists
and find its value.

If (xn) ∞ to n=1 is a convergent sequence with limn→∞ xn = 0
prove that
lim n→∞ (x1 + x2 + · · · + xn)/ n = 0 .

) Let α be a fixed positive real number, α > 0. For a
sequence {xn}, let x1 > √ α, and define x2, x3, x4, · · · by the
following recurrence relation xn+1 = 1 2 xn + α xn (a) Prove that
{xn} decreases monotonically (in other words, xn+1 − xn ≤ 0 for all
n). (b) Prove that {xn} is bounded from below. (Hint: use proof by
induction to show xn > √ α for all...

Define a sequence (xn)n≥1 recursively by x1 = 1 and
xn = 1 + 1 /(xn−1) for n > 0. Prove that limn→∞ xn = x exists
and find its value.

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 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) is a sequence of nonzero real numbers and if limn→∞ xn =
x where x does not equal zero; prove that lim n→∞ 1/ xn = 1/x

Let {xn} be a non-decreasing sequence and assume that xn goes to
x as n goes to infinity. Show that for all, n in N (naturals), xn
< x. Formulate and prove an analogous result for a
non-increasing sequences.

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.)

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.

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