1. ## Elementary Analysis

I have this proof class that is really bugging me with problems like these, I'd appreciate if someone could help me please.

(a) Let (Sn) be a sequence such that |Sn+1 - Sn|<2^-n for all n in N

Prove that (Sn) is a Cauchy sequence and hence a convergent sequence.

(b) Is the result in (a) true if we only assume that |Sn+1 - Sn|<1/n for all n in N?

2. Originally Posted by Jhevon
I have this proof class that is really bugging me with problems like these, I'd appreciate if someone could help me please.

(a) Let (Sn) be a sequence such that |Sn+1 - Sn|<2^-n for all n in N

Prove that (Sn) is a Cauchy sequence and hence a convergent sequence.
For m>n consider:

|Sm - Sn| = |(Sm - Sm-1) + (Sm-1 - Sm-2) + ... + (Sn+1 - Sn)|

................<= |Sm - Sm-1| + |Sm-1 - Sm-2| + ... + |Sn+1 - Sn|

................< 2^-(m-1) + 2^-(m-2) + ... + 2^-n

.................= 2^-n { 1 + 2^-1 + 2^-2 + ... 2^-(m-n) }

.................= 2^-n (1 - 2^[-(m-n)+1]) / (1-2^-1)

..................< 2^(-n+1)

hence as m, n go to infinity |Sm-Sn| -> 0, that is (Sn) is a Cauchy
sequence and hence converges.

(b) Is the result in (a) true if we only assume that |Sn+1 - Sn|<1/n for all n in N?
No, consider the partial sums of the harmonic series.

Sn = sum 1/r , r=1 .. n

We know that the harmonic series diverges hence (Sn) diverges, but:

|Sn+1 - Sn|= 1/(n+1) < 1/n.

Which provides the required counter example.

RonL

3. Originally Posted by Jhevon
I have this proof class that is really bugging me with problems like these, I'd appreciate if someone could help me please.

(a) Let (Sn) be a sequence such that |Sn+1 - Sn|<2^-n for all n in N

Prove that (Sn) is a Cauchy sequence and hence a convergent sequence.

(b) Is the result in (a) true if we only assume that |Sn+1 - Sn|<1/n for all n in N?
Are you using, Kenneth Ross book?
Chapter 11 on Cauchy Sequences and Monotone Sequences.?

4. Yup, that's the one.

I think it was chapter 10 though

5. Originally Posted by Jhevon
I think it was chapter 10 though
Are you by an chance in CCNY?
Because in the profile it says you are from NY.

6. Yes, I am from CCNY. I'm guessing you are as well, what math class are you in?

7. Originally Posted by Jhevon
Yes, I am from CCNY. I'm guessing you are as well, what math class are you in?
That is so cool. We are in the same class. MATH 323.
I guess I see you tomorrow.

8. Cool. How's the rest of the hw going?

9. Originally Posted by Jhevon
Cool. How's the rest of the hw going?
I finished it. I am a good boy, I do my homework as fast as I can instead of waiting to the last day.
(It is called Obsessive-Compulsive Disorder).

10. Nope, it's called being smart. Not only are you smart enough to finish the hw, but your free from the stress i'm going through. i'm worried to death about our test next week