Find the limit of a recursive sequence

[k3k3]

Homework Statement

Suppose |r|<1

Find the limit of {|r|^n} by treating it as a recursive sequence defined by x_1=|r| and x_n=|r|*x_(n-1)

Homework Equations

This is proving a theorem in the book:

If |r| < 1, then the sequence {r^n} converges to 0.

The Attempt at a Solution

It is clear that without the theorem this sequence converges to 0. Showing it is the issue, so maybe it is not so clear?

This is part 2, in part 1 I showed it converges by the monotone convergence theorem, so the limit does exist.

Suppose x_n -> L for some L in ℝ.

Since $lim_{n->∞}$x_n = L, then $lim_{n->∞}$x_n+$lim_{n->∞}$x_(n+1) = 2L

Which this is the same as saying $lim_{n->∞}$x_n+|r| $lim_{n->∞}$x_n = 2L

=> L+|r|L = 2L => L+|r|L-2L = 0 => -L+|r|L = 0 => L(-1+|r|)= 0 => L = 0

So the sequence converges to 0.


Am I correct or did I do something wrong?

 

[mighty2000]

Your solution is correct! You have correctly applied the recursive definition of the sequence and used the fact that |r|<1 to show that the limit must be 0. Well done! In part 1, you have also correctly used the monotone convergence theorem to show that the sequence is convergent. Keep up the good work!