56 Chapter 2 • Sequences. 3
By the Arch1medean property (remember that?), :3no EN 3 no> - + 3.
€
Take no= such a natural number.^4 The above analysis shows that
I
n > no :::;. ---2n +^3 - -^21 < c.
- 3n - 7 3
D
(
Example 2.1.6 Prove that lim 2n - + 3)^2
3
-- = -
3.
n-->oo n - 7Note: To prove this limit statement, we essentially work Part ( c) of Example
2.1.5 backwards. The work we did there is best regarded as "scratchwork" for
the proof we are about to give. We may include it as needed in our proof. But
keep in mind that our proof must stand alone. For that reason, whatever we
need from Example 2.1.5 (c) must be redone here.
Pay careful attention to the proof given here, as it will serve as a paradigm
for the proofs you will be required to give.
Let c > 0. By the Archimedean property, :3 no E N 3 no > ~ + 3. Then,
€
3
n ;::: no :::;. n > -+ 3 and n > 3
€3
:::;. n - 3 > - and n - 3 > 0
€n - 3 1
:::;. --> - and n - 3 > 0
3 €3
:::;. --< c (by Theorem 1.2.10, and since n - 3 > 0)
n-327
:::;. 9n - 27 <^623
:::;. < €.
9n - 21(Since - 21 > -27:::;. 9n - 21 > 9n - 27:::;.1
<1
)
9n - 21 9n - 27
and since 23 < 27.- Note that no depends on i::. There is no no such that no > ~ + 3 for all i:: > 0.