MthT 430 Notes Chap7c Bounded Monotone Sequences Have Limits
BISHL: Bounded Increasing Sequences Have Limits
Theorem. Let {xn}∞
n=1 be a bounded monotone increasing sequence; i.e.
x1≤x2≤. . . ,
and there is a number Msuch that for n= 1,2, . . .,
xn≤M.
Then there is a number Lsuch that
lim
n→∞
xn=L.
Proof using (P13–BIN): Without loss of generality, we assume that
0≤x1≤x2≤... ≤xn≤. . . < 1.
We will construct a binary expansion for L.
A picture is helpful!
Divide the interval [0,1) into two halves.
Is there an n1such that xn1≥m0=1
2= 0.bin1?
If NO, let c1= 0, a1= 0 = 0.binc1,b1=m0=a1+1
2. If YES, let c1= 1,
a1=m0=1
2= 0.binc1,b1=a1+1
2= 1.
In both cases, for n≥n1,a1= 0.binc1≤xn≤b1=a1+1
21and b1−a1=1
21.
Next Divide the interval [a1,b 1) = ·0.binc1, a1+1
21¶into two halves.
Is there an n2> n1such that xn2≥m1=a1+1
22?
If NO, let c2= 0, a2=a1= 0.binc1c2,b2=m1=a2+1
22. If YES, let c2= 1,
a2=m1= 0.binc1c2,b2=b1=a2+1
22.
In both cases, for n≥n2,a2= 0.binc1c2≤xn< b2=a2+1
22and b2−a2=1
22.
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