How can i prove these two convergence theorem?

  • #1
1. If {an} and {bn} are convergent, then {an士bn} and {anbn} are also convergent

2. If {an} and {bn} are convergent and there exists a constant k>0 such that |bn| > k for all n=1, 2, ...., then {an/bn} is also convergen
 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47
Well, given that an converges to a and bn converges to b, you could "guess" the limits of the other series and then try to prove that they actually are the limits from the definition.

E.g. you can try to show that
[tex]\forall_{\epsilon > 0} \exists_{N_+} : n > N_+ \implies |(a_n + b_n) - L| < \epsilon[/tex]
where L = a + b is the postulated limit. Of course you already know that given such an [itex]\epsilon[/itex] you can find Na and Nb such that
[tex]n > N_a \implies |a_n - a| < \epsilon[/tex]
and
[tex]n > N_b \implies |b_n - b| < \epsilon[/tex]
 

Related Threads on How can i prove these two convergence theorem?

  • Last Post
Replies
13
Views
9K
Replies
3
Views
2K
  • Last Post
Replies
3
Views
7K
Replies
4
Views
2K
Replies
2
Views
1K
Replies
1
Views
2K
  • Last Post
Replies
1
Views
3K
Replies
15
Views
7K
Top