More on Limit of a Sequence

Some basic properties.

1.

The limit of a convergent sequence is unique.

2.

Every convergent sequence is bounded.
This is a quite interesting result since it implies that if a sequence is not bounded, it is therefore divergent. For example, the sequence is not bounded, therefore it is divergent.

3.

Any bounded increasing (or decreasing) sequence is convergent.
Note that if the sequence is increasing (resp. decreasing), then the limit is the least-upper bound (resp. greatest-lower bound) of the numbers , for .

4.

If the sequences and are convergent and and are two arbitrary real numbers, then the new sequence is convergent. Moreover, we have

.

It is also true that the sequence is convergent and

.

5.

If the sequence is convergent and and for any , then the sequence is convergent. Moreover, we have

.

The following examples will be useful to familiarize yourself with limit of sequences.

Example: Show that for any number a such that 0 < a <1, we have

.

Answer: Since 0 < a <1, then the sequence is obviously decreasing and bounded; hence it is convergent. Write

.

We need to show that L=0. We have

,

since the sequence is a tail of the sequence ; hence they have the same limit. But,

using the previous properties, we get

,

which implies

.

Since , then we must have L=0.

One may wonder, what happened to the sequence if a > 1? It is divergent since it is not bounded. This follows from

and

.

Remark: Note that it is possible to talk about a sequence of numbers which converges to . Of course, we do reserve the word convergent to sequences which converges to a number; is not a number. The following shows the process:

The sequence converges to (or, to ), if and only if, for any real number M > 0, there exists an integer , such that
for any , we have (or ).

In particular, if as and for any , then we have

A sequence which converges to is obviously not bounded.
For example, we have

for any a > 1.

Example: Show that the sequence

is convergent.

Answer: Note that

Hence, we have

.

So the sequence is bounded. Next, let n=1, we get

and if we let n=2, we get

You may convince yourself by computing the first numbers of the sequence so that they are decreasing. It is natural to check whether this is the case, so we need to compare the two numbers

,

for any . We have

.

Since

and

we get

,

which implies

.

This is exactly what we expected. Therefore, the sequence is decreasing. Since it is bounded, we conclude that it is convergent.
The next natural question to ask is: what is the limit of the sequence? In the subsequent results we will show that in fact this sequence converges to 5.

Example: Show that

Answer: Since

then

.

Also, it is clear that for any . Putting the two results together, we conclude that is decreasing and bounded. Therefore, it is convergent. Let us write

.

We have , for any . Assume that . Then we have

which implies

for any . This is impossible since the sequence is not bounded (since ). Therefore, we have L=1, that is,

In fact, we have

for any a > 0.

The next result is extremely useful. It is known as the Pinching Theorem or the Sandwich Theorem:

Suppose that for a large n, we have

.

Assume that the sequences and are convergent and

.

Then the sequence is convergent. Moreover, we have

.

Example: Show that

.

Answer: We have

In the above examples, we showed that

The above theorem implies the conclusion

.

Example: Discuss the convergence of

Answer: It is not an easy example. The reason is that behaves very badly when n gets large. But we will make use of the fact that is bounded and gets small when n gets large. We have

.

Since

then by the Pinching Theorem, we conclude that

.

This is a very simple example which shows how powerful the Pinching Theorem is.

The next result is also useful and very commonly used.

Suppose that f(x) is a continuous function. If belongs to the domain of f(x) and , as , then we have

Example: Since , we have

a result we already proved in a harder way....

Example: Show that the sequence

is convergent and find its limit.
What do you think about

for any a > 0?

Next we will discuss some special limits.

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