Math Is Fun Forum

Infinite digits is tricky. The only way to deny 3.999...=4 is to say infinite digits and infinite division is illegal, meaningless or unpratical.

Humans can set up a real axis, but nature wouldn't bother to do the trick.

For What?

How did you deal with errors accured while construction?

The best game I think is Germany/Italy. And that's why I support Italy(Italia).

Portugal vs Netherlands, a game to remember.

Perhaps a Kongfu fight to remember~ :lol

Very good proof, Ricky. The main point is to admit a Maximum lower bound, then transform its property to N-∈ doctorine. I seem to get a little bit clue about why continuous reals is necessary for an ideal math world. 

By the way, what does s mean? you can simply say that you can find term n less than L+∈ which is not a lower bound.
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To mikau

Defining infinity and infinitesimal to a number is first invented by mathematician Thomson, and is still controversial because rules and properties have to alter to welcome their joining in numbers. Some mathematician just say this step is unnecessary.
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If I let a = n, as you do above, then it's the limit of n/n, and so it's 1.  But if I let a = 2n, which by your reasoning is also valid, then the limit is 2.  I could keep going, so there are infinitely different limits for the same problem?  Surely that's absurd.

--That's what I've insisted and you've admitted-reached state of infinity is artificial and can't live without moving cause.

Great proof in Latex! Your proof of

Good effort,Ricky!

Considering seri monotunely decreasing and bounded below has a limit, if your proof does not require a property of real numbers I would like to see it.

To Mikau, that is not a fair game, even when you choose 0.999999999 as a, n can be large enough to lower a^n down to 0.0000000001- the defination of limit is powerful enough to beat you.

Yes you got it! If you mean the rate of a water flow, or an electric current, or a ray of light, I can definitely tell you they are concrete and the rate  calculated from calculus is just an approximation.

Can you draw a real circle on the paper? I guess you cannot. Nevertheless, the formula with π discribes its area well. And it's not coincidence because a mass polynomials is just similar to an ideal circle.

The only two subjects so far people haven't discovered its discountinousity dispite their previous thought are time and space. But I don't know if space know a real axis.

Many years ago a Chinese mathematician said one statement:
" Let there be a pole, and cut the pole by half each day, you'll not get it eliminated even after millions of years."

This is used as the proof that we Chinese had the concept of limit.

But actually it is more like a statement discussing the flaw of limit, that you can approach the limit as close as you want, but you may never reach it in finite steps.

Zeno(n) suggests the same paradox, and he put it further, he said a moving arrow cannot move.

If you cannot add the series 0.5+0.25+0.125+...to zero, you will not get a 1, but the zero term does not show in term N when N is a natural number.

So does the series 0.5+0.25+0.125+...(infinite terms) make sense? does the notation 0.999...(infinite digits) make sense?

Why are we so addicted to get a perfect and precise result? Why don't we simply admit that infinity only exists in imaginary and reality is compatable with small residual errors?

By the way, none of a physic law envolving Calculus is universal, especially in small world, as far as I know.

Great that you like it. It's not always easy to find such an approprate metaphor because a structure similar stuff does not always exist.

for x, x>0 and x<1, x^n converges to 0

Proof of
for any given number ∈>0 there exist a number N such that n larger than N would give x^n<∈

x^N<∈ requires N<log(x,∈) hence such N exists.

proven.

But it's not the proof I previously referred.