SoyLeche
meh...
Hey, Ryan.
I've been trying to work through a more rigorous proof, but my skills at proving things mathematically are pretty rough (I've always been more of an application guy).
Here's what I want to prove (it's been seen here before):
x < y if and only if there exists a w such that x < w < y
Those are strict < signs, not <= signs, just to make sure there is no confusion. I have no doubt that this statement is true, but can it be proven?
Applying that to this situation we have:
.9999... < 1 if and only if there exists a w such that .9999... < w < 1
A little manipulation of the inequality leads to 0 < w - .9999... < 1 - .9999...
If Halcyon is to be believed, 1 - .9999... = .0000... which is greater than 1, so we have (set z = w - .9999...): 0 < z < .0000...
The next step, once the first proof is in place is to prove that w and z cannot exist. The proof for either is sufficient.
Can you help me with this? I asked my wife, who took some higher math classes than I did, and did a bit more on the theoretical side than me, but she said if I'd have asked her 3 years ago she probably could have done it. Now she's too rusty
I've been trying to work through a more rigorous proof, but my skills at proving things mathematically are pretty rough (I've always been more of an application guy).
Here's what I want to prove (it's been seen here before):
x < y if and only if there exists a w such that x < w < y
Those are strict < signs, not <= signs, just to make sure there is no confusion. I have no doubt that this statement is true, but can it be proven?
Applying that to this situation we have:
.9999... < 1 if and only if there exists a w such that .9999... < w < 1
A little manipulation of the inequality leads to 0 < w - .9999... < 1 - .9999...
If Halcyon is to be believed, 1 - .9999... = .0000... which is greater than 1, so we have (set z = w - .9999...): 0 < z < .0000...
The next step, once the first proof is in place is to prove that w and z cannot exist. The proof for either is sufficient.
Can you help me with this? I asked my wife, who took some higher math classes than I did, and did a bit more on the theoretical side than me, but she said if I'd have asked her 3 years ago she probably could have done it. Now she's too rusty