Proving Cauchy Sequences Using the Definition

[Mcoulombe]

Homework Statement

Im trying to prove that a sequence is cauchy using its definition. the sequence is 81+14n/50+31n. i know it converges to 14/31 using L'hopitals rule but the assignment is to use the definition of cauchy


i have tried some things but I am not 100% sure how to start all i have done it look at what

s_n - s_m in absolute value is

 

[micromass]

So what is |s_n-s_m|?

 

[Mcoulombe]

i got 1811m-1811n/2500+1550m+1550n+961mn

 

[Mcoulombe]

there is something else in also confused about the def i was given is:

a Sequence of real numbers, {s_n} is said to be cauchy if for each epsilon >0 there exists a real number N such that for all n,m in the set of natural numbers, we have:
n,m> N implies that S_n-s_m< epsilon.

what is N supposed to be and do i need to figure it out and i am no sure what epsilon is

 

[micromass]

You'll need to take an arbitrary (but fixed) real number $$\epsilon$$. For that specific epsilon, you need to find a suitable N such that

$$|s_n-s_m|<\epsilon$$

whenever n,m>N.

In this case, you'll need to show that

$$|1811m-1811n|<\epsilon|2500+1550m+1550n+961mn|$$

for large enough n and m.

For what follows, we can always assume (WLOG) that m>n. That way, we can leave out the absolute values. We need to show then that

$$1811m-1811n<\epsilon(2500+1550m+1550n+961mn)$$

for suitably large n,m.

 

[Mcoulombe]

1811m-1811n<e(2500+1550m+1550n+961mn)

so the LHS of this as m and n approach infinity is 0 obviously making the RHS larger. right?

 

[micromass]

Yes, except that the LHS does not approach 0. You've got a $$\infty-\infty$$ there. And that doesn't equal 0 necessairely...

 

[Mcoulombe]

ok i think i have done my proof but is all I am doing is showing
1811m-1811n<e(2500+1550m+1550n+961mn) or do i also need to find N from the definition

 

[micromass]

Well, you need to show that

1811m-1811n<e(2500+1550m+1550n+961mn)

but that won't hold for all n and m. It holds for all n and m that are greater then a certain N. You'll need to find that N...