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

Showing that an inequality is true

  • #1
1,456
44
Poster has been reminded that showing their work on schoolwork problems is mandatory at the PF

Homework Statement


Suppose that ##N \in \mathbb{N}## and that ##(s_n)## is a nonnegative sequence. Prove that ##\displaystyle \frac{s_{N+1} + s_{N+2} + \cdots + s_n}{n} \le \sup \{s_n ~:~ n > N \}##

Homework Equations




The Attempt at a Solution


I need help explaining why this is true. Supposedly it is obvious, but I can't quite see it...
 

Answers and Replies

  • #2
13,070
9,838
You have all ##s_k \le \mathcal{S} :=\sup\{s_n\, : \,n> N\}## for all ##k>N##.
Now count the summands on the left, each smaller than ##\mathcal{S}##, so ##M## many of them are smaller than?
 
  • Like
Likes nuuskur and Mr Davis 97
  • #3
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,543
755

Homework Statement


Suppose that ##N \in \mathbb{N}## and that ##(s_n)## is a nonnegative sequence. Prove that ##\displaystyle \frac{s_{N+1} + s_{N+2} + \cdots + s_n}{n} \le \sup \{s_n ~:~ n > N \}##

Homework Equations




The Attempt at a Solution


I need help explaining why this is true. Supposedly it is obvious, but I can't quite see it...
You have to show some effort. Once you do, you may see it....
 
  • #4
542
360
Well, if [itex]s_n[/itex] is a diverging sequence, the claim is trivially true. Suppose the sequence converges, then it's bounded. Can an arithmetic mean of some n consecutive elements in the sequence exceed that bound? Any bound (hint hint, the smallest bound), for that matter.

The elements can also be picked arbitrarily, they don't have to be consecutive. The claim will still hold.

One can generalise even further and state a similar claim for arbitrary sequences, not just nonnegative ones.
 

Related Threads on Showing that an inequality is true

Replies
3
Views
1K
Replies
2
Views
2K
Replies
5
Views
805
Replies
6
Views
892
Replies
5
Views
895
  • Last Post
Replies
1
Views
947
Replies
2
Views
2K
Replies
15
Views
2K
Replies
11
Views
2K
Top