This preview shows pages 1–3. Sign up to view the full content.
Pg 25, Proposition 2.1 – The sequence {1/n} converges to 0; that is,
Pf:
Let . Then by the definition of convergence there is a natural number N such that
Therefore,
when . By Corollary 1.10, we can say that there is a natural number N such that . So,
Therefore, the sequence {1/n} converges to 0.
QED
Pg 25, Example 2.5 – Prove that the sequence does not converge.
Pf:
For purposes of contradiction let’s consider the sequence does converge to a real number a. We will let . By
the definition of convergence there is a natural number N such that
We can see that , so .
But we also see that , so .
Therefore, the sequence does not converge.
QED
Pg 26, Example 2.6 – Prove that the sequence
converges to 3; that is, .
Pf:
Let . By the definition of convergence there is a natural number N such that
Notice that .
Notice that
is a positive number. By Corollary 1.10, there is a natural number N such that
or .
Therefore,
. So we have shown that sequence
converges to 3.
QED
Pg 26, Proposition 2.2 – For any number c such that c < 1, the sequence
converges to 0; that is,
Pf:
Let . By the definition of convergence there is a natural number N such that
So,
when . Let , then we can see that . Notice that d is a positive number, so by Bernoulli’s Inequality
Notice that d
ϵ
is a positive number. So by Corollary 1.10, there is an natural number N such that
or . So,
Therefore, when c < 1, the sequence
converges to 0.
QED
Pg 27, Lemma 2.3 – Every convergent sequence is bounded.
Pf:
Let
be a sequence that converges to the number a. Taking , it follows from the definition of convergence
that we may select a natural number N such that
By using the Reverse Triangle Inequality, we have
From which it follows that
Define . Then,
Therefore, every convergent sequence is bounded.
QED
Pg 27, Lemma 2.4 – Suppose that the sequence
converges to the nonzero number b. Then there is a natural
number N such that
Pf:
Notice that
is a positive number. So, let . By the definition of convergence there is a natural number N such
that
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document By the Triangle Inequality we have
With basic algebra can see that
QED
Pg 28, Theorem 2.5 – Suppose that the sequence
converges to the number a and that the sequence
converges
to the number b. Then the sequence
converges, and
Also, the sequence
converges, and
Moreover, if
for all n and , then the sequence
converges, and
Pf of (i):
Let . By the definition of convergence we need a natural number N such that
By the definition of convergence we know that there is a natural number N
1
such that
We also know that there is a natural number N
2
such that
Let . With the Triangle Inequality we see that
Therefore (i) is true.
QED
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 11/10/2011 for the course MATH 511 taught by Professor Higdon during the Spring '11 term at Oregon State.
 Spring '11
 Higdon

Click to edit the document details