Question

prove that every convergent sequence is bounded. ​

Answer 1

Step-by-step explanation:

Proof that Convergent Sequences are Bounded

We are now going to look at an important theorem - one that states that if a sequence is convergent, then the sequence is also bounded.

Theorem: If {an} is a convergent sequence, that is limn→∞an=L for some , then is also bounded, that is for some , .

Proof of Theorem: We first want to choose where such that . Choose to be some positive number for epsilon, let's say . Therefore and by the triangle inequality ():

(1)

So if , then .

Now consider where . This is a finite set so there exists a maximum value, call it , that is .

We will now summarize that if , then the maximum value the sequence takes on is . If , then the maximum value the sequence takes on is . Therefore, let . Therefore, , , so we have shown that is bounded