Determine whether the sequence with the given n'th term is monotonic and whether it is bounded.

Let write the first few terms of the sequence:

We can see that the first four terms are increasing so it is possible that the whole sequence is monotonically increasing. To prove that, we need to check that

Now we multiply the whole inequality by  We can do that because  and thus 

Since zero is indeed less than six we can...

Let write the first few terms of the sequence:

We can see that the first four terms are increasing so it is possible that the whole sequence is monotonically increasing. To prove that, we need to check that

Now we multiply the whole inequality by  We can do that because  and thus 

Since zero is indeed less than six we can conclude that Therefore, the sequence is monotonically increasing.

Because the sequence is monotonically increasing it is bounded from below i.e.

Let us now prove that the sequence is bounded from above by 3 i.e.

Now we multiply by  We can do that because  and thus 

Since zero is indeed less than six we can conclude that   Therefore, the sequence is bounded from both below and above i.e.

The image below shows first 50 terms of the sequence. Both monotonicity and boundedness can clearly be seen on the image.