Monotone and bounded sequence

  • #1
Let (xn) be a seq of real nos and let sn = x1+x2+x3+...+xn / n.

prove that if if xn is bounded and monotone, then sn is also bdd and monotone.


How can i got about this one.. ?

I got it in the test today and i couldn't figure it out. only hint i could think of is how do i prove if xn is increasing because if i prove tht i can prove it. but i could not do it

please some one suggest
 

Answers and Replies

  • #2
1,101
3
You're not supposed to prove that x_n is increasing since you're basically given that x_n is either increasing or decreasing. This is what monotone means, though perhaps you would replace increasing with non-decreasing and decreasing with non-increasing if you allowed subsequent terms in the sequence to be equal to previous terms.

You need to prove two things here: 1. s_n is bounded, 2. s_n is monotone. First, to show s_n is bounded, you obviously need to use the hypothesis that x_n is bounded. What does it mean for x_n to be bounded?
 

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