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 tex2html_wrap_inline193 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 tex2html_wrap_inline195 , for tex2html_wrap_inline197.
4.
If the sequences tex2html_wrap_inline199 and tex2html_wrap_inline201 are convergent and tex2html_wrap_inline203 and tex2html_wrap_inline205 are two arbitrary real numbers, then the new sequence tex2html_wrap_inline207 is convergent. Moreover, we have

displaymath209.

It is also true that the sequence tex2html_wrap_inline211 is convergent and

displaymath213.

5.
If the sequence tex2html_wrap_inline199 is convergent and tex2html_wrap_inline217 and tex2html_wrap_inline219 for any tex2html_wrap_inline221 , then the sequence tex2html_wrap_inline223 is convergent. Moreover, we have

displaymath225.

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

displaymath231.

Answer: Since 0 < a <1, then the sequence tex2html_wrap_inline235 is obviously decreasing and bounded; hence it is convergent. Write

displaymath237.

We need to show that L=0. We have

displaymath241,

since the sequence tex2html_wrap_inline243 is a tail of the sequence tex2html_wrap_inline235; hence they have the same limit. But,

displaymath247

using the previous properties, we get

displaymath249,

which implies

displaymath251.

Since tex2html_wrap_inline253 , then we must have L=0.

One may wonder, what happened to the sequence tex2html_wrap_inline235 if a > 1? It is divergent since it is not bounded. This follows from

displaymath261

and

displaymath263.

Remark: Note that it is possible to talk about a sequence of numbers which converges to tex2html_wrap_inline265 . Of course, we do reserve the word convergent to sequences which converges to a number; tex2html_wrap_inline265 is not a number. The following shows the process:

The sequence tex2html_wrap_inline199 converges to tex2html_wrap_inline271 (or, to tex2html_wrap_inline273 ), if and only if, for any real number M > 0, there exists an integer tex2html_wrap_inline277, such that
for any tex2html_wrap_inline279 , we have tex2html_wrap_inline281 (or tex2html_wrap_inline283 ).

In particular, if tex2html_wrap_inline285 as tex2html_wrap_inline287 and tex2html_wrap_inline289 for any tex2html_wrap_inline221 , then we have

displaymath293

A sequence which converges to tex2html_wrap_inline265 is obviously not bounded.
For example, we have

displaymath297

for any a > 1.

Example: Show that the sequence

displaymath301

is convergent.

Answer: Note that

displaymath303

Hence, we have

displaymath305.

So the sequence tex2html_wrap_inline307 is bounded. Next, let n=1, we get

displaymath311

and if we let n=2, we get

displaymath315

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

displaymath317,

for any tex2html_wrap_inline221. We have

displaymath321.

Since

displaymath323

and

displaymath325

we get

displaymath327,

which implies

displaymath329.

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

displaymath331

Answer: Since

displaymath333

then

displaymath335.

Also, it is clear that tex2html_wrap_inline337 for any tex2html_wrap_inline221 . Putting the two results together, we conclude that tex2html_wrap_inline341 is decreasing and bounded. Therefore, it is convergent. Let us write

displaymath343.

We have tex2html_wrap_inline345 , for any tex2html_wrap_inline221 . Assume that tex2html_wrap_inline349 . Then we have

displaymath351

which implies

displaymath353

for any tex2html_wrap_inline221 . This is impossible since the sequence tex2html_wrap_inline357 is not bounded (since tex2html_wrap_inline359 ). Therefore, we have L=1, that is,

displaymath331

In fact, we have

displaymath365

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

displaymath371.

Assume that the sequences tex2html_wrap_inline199 and tex2html_wrap_inline201 are convergent and

displaymath377.

Then the sequence tex2html_wrap_inline379 is convergent. Moreover, we have

displaymath381.

Example: Show that

displaymath383.

Answer: We have

displaymath385

In the above examples, we showed that

displaymath331

The above theorem implies the conclusion

displaymath383.

Example: Discuss the convergence of

displaymath391

Answer: It is not an easy example. The reason is that tex2html_wrap_inline393 behaves very badly when n gets large. But we will make use of the fact that tex2html_wrap_inline393 is bounded and tex2html_wrap_inline399 gets small when n gets large. We have

displaymath403.

Since

displaymath405

then by the Pinching Theorem, we conclude that

displaymath407.

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 tex2html_wrap_inline195 belongs to the domain of f(x) and tex2html_wrap_inline415 , as tex2html_wrap_inline287 , then we have

displaymath419

Example: Since tex2html_wrap_inline421 , we have

displaymath423

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

Example: Show that the sequence

displaymath425

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

displaymath427

for any a > 0?

Next we will discuss some special limits.

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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