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Unformatted text preview: s for Limits
1. Suppose that x n and y n are sequences in R k , that x n x as n Ý and that y is a partial limit of y n .
Prove that x y is a partial limit of the sequence x n y n .
Suppose that 0. Using the fact that x n converges to the point x we choose an integer N such
that the inequality
xn x
2
holds whenever n N. Since y is a partial limit of the sequence y n and since there are only
finitely many positive integers n N there must be infinitely many integers n N for which the
inequality
yn y
2
holds. For each of these infinitely many integers we have
xn yn x y xn x yn y
x yn xn y 2 2 . 2. Is it true that if x n and y n are sequences in R k and x is a partial limit of x n and that y is a partial limit of
y n then x y is a partial limit of the sequence x n y n ?
The answer is no; even in the metric space R.
Define x n y n 1 n for each n. We observe that the numbers 1 and 1 are partial limits of x n
and y n respectively but that 1 1 fails to be a partial limit of x n y n .
3. State and prove some analogues of Exercise 1 for differences and inner products in R k and for products and
quotients in R.
Suppose that x n and y n are sequences of numbers, that x n converges to a number x and that
a real number y is a partial limit of y n . We shall prove that the number xy must be a partial limit of
the sequence x n y n .
Using the fact that x n is convergent, and therefore bounded, we choose a number p such that
x n  p for every n. For each n we observe that
x n y n xy  x n y n x n y x n y xy 
x n y n
py n x n y  x n y
y  y x n xy 
x Now, to show that xy is a partial limit of the sequence x n y n , suppose that 0.
Using the
fact that x n x as n Ý we choose an integer N such that the inequality
x n x 
2y  1
holds whenever n N. Since there are only finitely many positive integers n N, and y is a partial
limit of the sequence y n , there must be infinitely many integers n N for which the inequality
y n y 
2p
holds. For each of these infinitely many integers n we have
x n y n xy  x n y n x n y x n y xy 
x n y n
py n
p 2p 164 x n y  x n y
y  y x n
y  xy 
x 2y  1 . 4. Suppose that x n and y n are sequences in R 3 that converge respectively to points x and y. Prove that
lim x
y n x y.
nÝ n
We write each point x n in the form
xn an, bn, cn
and each point y n in the form
yn un, vn, wn
and
x a, b, c
and
The given information tells us that as n
w n w. Therefore
yn n Ý bnwn
lim
lim x
nÝ n
bw cv, cu y u, v, w .
Ý we have a n a, b n cnvn, cnun anwn, anvn aw, av b, c n bu a, b, c c, u n u, v n v and bnun
u, v, w x y. 5. Give an example of two divergent sequences x n and y n in R such that the sequence x n y n is
convergent.
We define x n 1 n and y n 1 n 1 for each n. Note that the sequence x n y n is the sequence
with constant value 0.
6. Give an example of two sequences x n and y n in R such that x n 0 and y n Ý and a. x n y n 0
We...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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