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
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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.
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 Spring '11
 Higdon
 Order theory, Natural number, Limit of a sequence

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