• Support PF! Buy your school textbooks, materials and every day products Here!

Show that sequence is unbounded

  • Thread starter oceanthang
  • Start date
  • #1

Homework Statement


Show that the sequence an= sqrt(n) is unbounded


Homework Equations


there is no relevant equation require.


The Attempt at a Solution


Actually, I'm a newbie for real analysis, I try to prove it by using contradiction method,
but I stuck at half way, can anyone provide solution to me? thanks.
 

Answers and Replies

  • #2
9
0
Suppose an is bounded. Then there exists M > 0 such that an <= M for all n. However this is absurd since you only have to pick n greater than M^2 to have an > M.

To illustrate: pick M = 120. 120^2 = 14400. a_14401 > 120.

Makes sense?
 
  • #3
Can you explain more detail about the illustration? I don't really get it about this statement
"only have to pick n greater than M^2 to have an > M", thanks.
 
  • #4
133
1
Could you show us what you did to become stuck half way? We can assist you by checking if your approach was going the right way.
 
  • #5
9
0
Can you explain more detail about the illustration? I don't really get it about this statement
"only have to pick n greater than M^2 to have an > M", thanks.
Stating that a_n is bounded means that you can find a value of M which is greater than all the terms in the sequence. You must prove there is no such M. Reduction to the absurd works well in this situation.

You suppose there exists M >= a_n for all n. But there is a problem with this statement, which is a_{[M^2] + 1} > M, where [M^2] is the integer part of M squared.

Suppose you state that M = 120.2 is greater than all the terms in the sequence (a purely arbitrary choice).
That is absurd since a_{[120.2^2]+1} = a_14449 = sqrt(14449) > 120.2. This reasoning works for all real M.
 
  • #6
I finally get it, thank you~
 

Related Threads for: Show that sequence is unbounded

  • Last Post
Replies
4
Views
719
  • Last Post
Replies
7
Views
435
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
16
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
16
Views
10K
Replies
7
Views
520
  • Last Post
Replies
4
Views
2K
Top