Question

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.

Answer #1

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.

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:...

Let (X1, Y1), . . . ,(Xn, Yn), be a random sample from a
bivariate normal distribution with parameters µ1, µ2, σ2 1 , σ2 2 ,
ρ. (Note: (X1, Y1), . . . ,(Xn, Yn) are independent). What is the
joint distribution of (X ¯ , Y¯ )?

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 a sequence in a metric space X . If Xn -> x in X
iff every neighbourhood of x contains all but finitely many points
of the terms of {Xn}

) 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...

Find an example of a sequence, {xn}, that does
not converge, but has a convergent subsequence.
Explain why {xn} (the divergent sequence) must have an
infinite number of convergent subsequences.

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

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