# Limits Proof

1. Dec 10, 2008

### khdani

Hello,
given lim(an)=a and lim(bn)=b if a<b prove that an < bn.

can i say that if a/b < 1 than an<bn ?

2. Dec 10, 2008

### e(ho0n3

What if b = 0?

3. Dec 10, 2008

### khdani

if b=0 than a is negative and probably an < bn
so it still holds, but how do i prove that?

4. Dec 10, 2008

### e(ho0n3

5. Dec 10, 2008

### khdani

can I say that
lim(an-bn) = (a-b) < 0
hence an < bn ?

6. Dec 10, 2008

### e(ho0n3

You can, but it doesn't convince me that an < bn. Also, I think that what you're trying to prove is false.

7. Dec 10, 2008

### HallsofIvy

Staff Emeritus
You can't prove it- it isn't true. What you can prove is that for n large enough, an< bn. Use the definition of limit with $\epsilon$ less that half the difference between a and b.

But you cannot say anything about an and bn for smaller values of n.

No, that's not true either. Again, it is only true for "sufficiently large" n.

Got example, an= 1/n converges to 1 while bn= 1/2n for n= 1 to 1000000, bn= 2- 1/n for n> 1000000 converges to 2 (so a= 0< 2= b and a/b= 1/2< 1) but an< bn only for n> 1000000. And you should be able to see how to make examples where that is true only for n> whatever number you want.