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Finding
N
: a vague outline
Math 8 2009, Chris Lyons
Let’s suppose you have a sequence
{
a
n
}
of real numbers, and you’re trying to prove that
lim
n
→
∞
a
n
=
L
by using the deﬁnition of the limit. Your proof reads as follows:
Proof.
Suppose that
ε >
0 is given. Choose
N
to be
... and that’s as far as you’ve gotten.
How do you ﬁnd the right
N
?
In general, this is a tough question to answer and it takes practice to ﬁnd the right
N
in diﬀerent
situations. But here is a vague strategy on how to proceed. (You can also see the example below, which
ought to help clarify the meaning of the steps.)
1. Take the quantity

a
n

L

and try to simplify it as much as possible. If you’re able to remove the
absolute value signs somehow, that also helps.
2. Try to ﬁnd some (positive) function
g
(
n
) which has both of the following properties:
(a) It’s
easy
to show that there exists some
N
1
such that
n
≥
N
1
=
⇒
g
(
n
)
< ε.
(b) There exists some
N
2
such that
n
≥
N
2
=
⇒ 
a
n

L
 ≤
g
(
n
)
.
3. Choose
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.
 Spring '08
 Ramakrishnan
 Real Numbers

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