Proving Convergence and Limit of a Sequence (Xn) in R

[cb1020102022]

Homework Statement

let (Xn) be a sequence in R. Let (An) be a sequence defined as An=(X1 +X2+...Xn-1+Xn)/n. (Xn) is a convergent sequence and the limit of Xn as n goes to infinity is L. Prove (An) in convergent sequence and that the limit is also L.

Homework Equations

The Attempt at a Solution

 

[SammyS]

What have you tried? Where are you stuck.

Write out the first few terms of (A_n).

 

[cb1020102022]

A1=X1, A2= (X1+X2)/2, A3=(X1+X2+X3)/3... I HAVE A THM IN MY BOOK THAT SAYS IF THE SUMMATION OF A SEQUENCE Xn CONVERGES THEN THE LIMIT OF Xn is 0. So I first assumed that the limit of Xn is not 0 the the numerator of An does not converge. But that does not really help. I am trying to figure out which convergence theorem to use...

 

[SammyS]

So the n-th term, A_n, is the average of the first n terms of sequence (X_n).

 

[cb1020102022]

right! But I am still stuck...

 

[Dick]

right! But I am still stuck...

You need to put together an argument using the definition of limit here. Pick an epsilon, how would find an N such that the average of the terms up to k is within epsilon of L for all k>N? The initial terms in the sequence may not be close to L, the later ones have to be. You need to take both groups into account.