# Basic Examples

Proofs for the problems below follow a common structure, as described on the prior page.
We strongly encourage readers attempt each problem by first drawing a picture, and then attempting a formal write-up before reading the example write-up provided below.
Example 1.
Define the sequence
$\{a_n\}$
by
$a_n \triangleq \dfrac{n}{n^2+9}$
for all
$n\in\mathbb{N}.$
Prove
$\{a_n\}$
converges to zero.
Ex 1: Comment
Hint
Visual
Formal Write-Up
Follow-Up
Although the result may be clear from calculus, the point here is to provide formal arguments.
Note
$0 < a_n < \dfrac{1}{n}.$ Plot of and . We see , and so as . Also note for large .
We claim
$\{a_n\}$
converges to zero and prove this as follows. Let
$\varepsilon > 0$
be given. It suffices to show there exists
$N\in\mathbb{N}$
such that
$|a_n - 0 | \leq\varepsilon$
for all
$n\geq N.$
For each
$n\in \mathbb{N}$
, note
$n > 0$
and so
$1/n > 0$
, which implies
 ​$n < n + \dfrac{9}{n} \ \ \implies \ \ 1 < \dfrac{n + \frac{9}{n}}{n} \ \ \implies \ \ \dfrac{1}{n + \frac{9}{n}} < \dfrac{1}{n}.$​
Consequently,
 ​​
where the second equality holds since
$n > 0$
and
$n^2 + 9 > 0$
. By the Archimedean Property of
$\mathbb{R}$
, there exists
$\tilde{N}\in\mathbb{N}$
such that
$1 < \frac{\tilde{N}}{\varepsilon}$
, which implies
$\frac{1}{\tilde{N}}< \varepsilon$
. Thus, combining the above results,
 ​​
which verifies the desired inequality, taking
$N=\tilde{N}.$
 ​$\blacksquare$​
• We emphasize each statement is a complete sentence (creativity is not required).
• It is completely acceptable to mirror this structure, but change the inequalities/terms for another problem.
• Readability is greatly aided by concluding with a statement that explicitly states how an initial task (which we set out to do) was completed.
Example 2.
Define the sequence
$\{a_n\}$
​ by
$\dfrac{4n^3+n}{n^3+6}$
​ for all
$n\in\mathbb{N}.$
​ Prove
$\{a_n\}$
converges.​
Ex 2: Comment
Hint
Visual
Formal Write-Up
We can show
$\{a_n\}$
​ converges by verifying it converges to a limit (that you should find).
As
$n$
​ gets big, the dominant term in the numerator is
$4n^3$
​ and in the denominator is
$n^3$
​, which should give insight about the limit of
$\{a_n\}.$ Animation of the sequence converging.
We claim
$\{a_n\}$
​ converges to four and prove this as follows. Let
$\varepsilon>0$
​ be given. If suffices to show there is
$N\in\mathbb{N}$
​ such that
$|a_n-0|\leq\varepsilon$
​ for all
$n\geq N.$
​ For each
$n\in\mathbb{N}$
​, observe
 ​$|a_n-4|=\left|\dfrac{4n^3+n}{n^3+6}-4\right|=\left| \dfrac{(4n^3+n)-(n^3+6)4}{n^3+6}\right|=\left|\dfrac{n-24}{n^3+6}\right|.$​
Building on this inequality,
$n\geq 24$
​ implies
 ​$|a_n - 4| = \dfrac{n-24}{n^3+6} \leq \dfrac{n}{n^3+6} \leq \dfrac{n}{n^3} \leq \dfrac{n}{n^2}=\dfrac{1}{n},$​
​where we note
$n^2=n\cdot n \geq n\cdot 1=n$
​ implies
$1/n^2\leq 1/n.$
​ By the Archimedean property of
$\mathbb{R}$
​, there exists a natural number
$N_1\in\mathbb{N}$
​ such that
$1 \leq N_1 \varepsilon$
​, and so
$1/N_1 \leq \varepsilon$
​. Setting
$N_2 = \max(24,\ N_1)$
yields, by the above results,
 ​​
This verifies the desired inequality, taking
$N=N_2.$
 ​$\blacksquare$​
Example 3.
Suppose
$\{c_n\}$
​ is a sequence in
$[-3,7)$
, i.e.
$-1\leq c_n<7$
for all
$n\in\mathbb{N}.$
​ Define the sequence
$\{a_n\}$
​ by
$a_n\triangleq c_n/n$
​ for all
$n\in\mathbb{N}.$
​ Prove
$\lim_{n\rightarrow\infty} a_n = 0.$
Ex 3: Comment
Hint
Visual
Formal Write-Up
Follow-Up
Be careful to formally state what must be shown (to ensure no confusion between the use of each of the listed sequences).
Here is it helpful to bound
$a_n$
​ in terms of a bound for
$c_n$
​. Example sequence convergence within bounds for 7/n and -7/n.
Let
$\varepsilon > 0$
​ be given. It suffices to show there is
$N\in\mathbb{N}$
​ such that
$|a_n - 0|\leq \varepsilon$
​ for all
$n\geq N.$
​ By our hypothesis on
$\{c_n\}$
​,
 ​​
By the Archimedean property of
$\mathbb{R}$
​, there is
$N_1\in\mathbb{N}$
​ such that
$7 < N_1 \varepsilon$
​, which implies
$7/N_1 < \varepsilon.$
​ Hence
 ​​
This shows the desired result, taking
$N=N_1$
​.
 ​$\blacksquare$​
A trend in the previous three problems is to use the Archimedean property of
$\mathbb{R}$
​. This is not a coincidence! Be sure to know how to use this property so you may obtain the existence of the natural number needed in the definition of limit convergence.

### Exercises.

Exercise 1. Prove the sequence
$\{a_n\}$
defined by
$a_n\triangleq \dfrac{2n^2+7}{3+n^2}$
converges to two.
• State the definition of what must be shown.
• Rewrite the inequality with
$a_n - 2.$
• Apply the Archimedean property of
$\mathbb{R}$
.
• Combine relations to verify the inequality that must be shown.​