Prove that the sequence defined by X1=3 and xn1 
1
4 xn
converges
Proof. We use the following theorem to prove it.
Theorem. If {x n } is a decreasing sequence and there exists a constant C such that for all
n  N , xn  C , then the sequence {x n } converges.
First, we need to show that n  N , 3  xn  0 . We prove it by induction.
Since 3  x1  3  0 , we know that it is true for n=1. Assume that it is true for n, namely,
3  xn  0 . We will prove it is true for n+1, namely, 3  xn1  0 .
Since xn1  41xn , we have
0  14 
1
4 0
 xn1 
1
4 xn

1
43
1 3
which implies that 3  xn1  0 . So, we conclude that for all n  N ,
(*)
3  xn  0
Now we will show that {x n } is a decreasing sequence, namely, xn  xn1
We prove it by induction as well. Since x1  3 and xn1  41xn , we get
x2 
1
4 x1

1
43
 1  x1 . So, xn  xn1 is true for n=1. Assume that it is true for n, namely,
xn  xn1 , we will show that it is true for n+1, namely, xn 1  x n  2
We know that
x n 1  x n  2 
1
4  xn
 4 1xn 1 
x n  x n 1
( 4  xn )( 4  xn 1 )
(1)
By hypothesis of induction, we have xn  xn1  xn  xn1  0 . By the fact
(*): 3  xn  0 and (1) , we can obtain
x n 1  x n  2 
1
4  xn
 4 1xn 1 
x n  x n 1
( 4  xn )( 4  xn 1 )
0
So, we have proved that xn  xn1 for all n  N .
From above, we have shown that {x n } is a decreasing sequence and there exists a
constant C=0 such that for all n  N , xn  0 . So by theorem, we are done.