Another sequence convergence proof

  • Thread starter antiemptyv
  • Start date
  • #1
34
0

Homework Statement



Let [tex]y_n := \sqrt{n+1} - \sqrt{n}[/tex] for [tex]n \in \mathbb{N}[/tex]. Show that [tex] (y_n)[/tex] converges.

Homework Equations



The Attempt at a Solution



I see that it converges to 0. I just need a nudge in the right direction at getting into [tex]| \sqrt{n+1} - \sqrt{n} - 0 | = | \sqrt{n+1} - \sqrt{n} |[/tex] to show it's less than any [tex]\epsilon > 0[/tex]. Any manipulating I've tried so far makes the terms way too big to work with.
 
Last edited:

Answers and Replies

  • #2
morphism
Science Advisor
Homework Helper
2,015
4
How about

[tex]\left(\sqrt{n+1} - \sqrt{n}\right) \cdot \frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}}[/tex]
 
  • #3
34
0
ohhhh, i see it now.
 
  • #4
4
0
What do you do after
1/(sqrt{n+1)+sqrt{n}) ??
 

Related Threads on Another sequence convergence proof

  • Last Post
Replies
3
Views
886
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
859
  • Last Post
Replies
2
Views
932
  • Last Post
Replies
14
Views
1K
  • Last Post
Replies
1
Views
830
  • Last Post
Replies
2
Views
985
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
2K
Top