Basic Examples
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
by
for all
Prove
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

Plot of and . We see , and so as . Also note for large .
We claim
converges to zero and prove this as follows. Let
be given. It suffices to show there exists
such that
for all
For each
, note
and so
, which implies
|
Consequently,
|
where the second equality holds since
and
. By the Archimedean Property of
, there exists
such that
, which implies
. Thus, combining the above results,
|
which verifies the desired inequality, taking
|
- 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
by
for all
Prove
converges.
Ex 2: Comment
Hint
Visual
Formal Write-Up
We can show
converges by verifying it converges to a limit (that you should find).
As
gets big, the dominant term in the numerator is
and in the denominator is
, which should give insight about the limit of

Animation of the sequence converging.
We claim
converges to four and prove this as follows. Let
be given. If suffices to show there is
such that
for all
For each
, observe
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Building on this inequality,
implies
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where we note
implies
By the Archimedean property of
, there exists a natural number
such that
, and so
. Setting
yields, by the above results,
|
This verifies the desired inequality, taking
|
Example 3.
Suppose
is a sequence in
, i.e.
for all
Define the sequence
by
for all
Prove
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
in terms of a bound for
.
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Example sequence convergence within bounds for 7/n and -7/n.
Let
be given. It suffices to show there is
such that
for all
By our hypothesis on
,
|
By the Archimedean property of
, there is
such that
, which implies
Hence
|
This shows the desired result, taking
.
|
A trend in the previous three problems is to use the Archimedean property of
. 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.
Exercise 1. Prove the sequence
defined by
converges to two.
Last modified 1mo ago