Could somebody validate my proof regarding the limit of when, tends to ?
So, let me cearly state the problem:
Let be a convergent sequence, with: , , n natural number, and , with . Then
Here is my idea for a proof:
Our goal is to proof that there there is some , such that , we have that
So here is what I did. First:
Then, beacause , , it easily follows that , . Applying this, we have that:
Now, we use the basic property of the absolute value: , that gives us:
Now, we use the fact that . So, there is some , such that , we have that . We choose an epsilon that takes the form . This choice is possible for any
Now, we have managed to obtain that:
Since we can find an , such that , the above inequality is staisfied, our claim is proved.
So, can you please tell me if my proof si correct? I've tried to find a proof, only for the limit of sequences! This problem has been on my nerves for s while. Also, probably there is some simpler way to do it, but I couldn't find it.