Mathematical Proof that real numbers are Finite.

Proof that all numbers between 0 to 1 as fractions as 0.1, 0.2, 0.3........The total of real positive numbers to infinity are the same as counting 1,2,3,4...... The only difference is that 0.X on the line between 0 to 1 line  is actually equal to the number X on the real number line.Thus since all finally end with 1 ,the total of real positive numbers isn't infinite as we thought but finite.

What we are describing is known as the one-to-one correspondence between the interval [0,1] and the set of all real numbers.

To prove that the two sets have the same cardinality (i.e., the same number of elements), we need to show that there exists a bijective function between them. In other words, we need to find a function that maps each element of the interval [0,1] to a unique element of the set of all real numbers, and vice versa.One such function is the following:

f(x) = x/(1-x)

This function maps every number x in the interval [0,1) to a unique number f(x) in the set of all real numbers, and it maps 1 to infinity.

To prove that f(x) is bijective, you need to show that it is both injective (i.e., each element in the domain maps to a unique element in the codomain) and surjective (i.e., every element in the codomain is mapped to by at least one element in the domain).

Injectivity: Suppose f(x1) = f(x2) for some x1, x2 in the domain of f. Then we have x1/(1-x1) = x2/(1-x2). Cross-multiplying and simplifying, we get x1 = x2. Therefore, f is injective.

Surjectivity: Let y be any real number. If y is positive, then y = x/(1-x) for some x in the interval [0,1). Solving for x, we get x = y/(y+1), which is in the domain of f. If y is negative, then we can choose x = y/(1-y), which is also in the domain of f and maps to y. Finally, if y = 0, we have f(0) = 0. Therefore, f is surjective.

Since f is both injective and surjective, it is bijective, and we have shown that the interval [0,1] and the set of all real numbers have the same cardinality.

In summary since all number fractions between 0----1 finally end at number 1and the left side of the equation is equal to the right which represents the real numbers.it proves that all positive real numbers are finite and not infinite.

Image of numbers 

Now I have several proofs that I don't know which is right and which is wrong . 🤔 🤔

Proof two question;

So it's not possible to prove a mathematical relationship between all the possible fractions between 0 and 1on a number line and map them onto all the real numbers btn 0 and infinity on a real numbers line .

Answer.

Actually, it is possible to prove that there is a one-to-one correspondence between the set of all fractions between 0 and 1 and the set of all real numbers between 0 and infinity. In fact, this result is known as Cantor's diagonal argument, which is a proof technique used in set theory.

The proof works by assuming that there is no one-to-one correspondence between the two sets and then constructing a mapping that contradicts this assumption. Here is a brief sketch of the proof:

Assume that there is no one-to-one correspondence between the set of all fractions between 0 and 1 and the set of all real numbers between 0 and infinity. Then there must be some real numbers between 0 and infinity that do not have a corresponding fraction between 0 and 1.

Let's represent each fraction between 0 and 1 as an infinite sequence of digits after the decimal point. For example, 1/3 is represented as 0.3333..., 1/7 is represented as 0.142857142857..., and so on.

Now, we construct a new number by taking the first digit after the decimal point of the first fraction, the second digit after the decimal point of the second fraction, the third digit after the decimal point of the third fraction, and so on. We then add 1 to each digit (mod 10), so that we get a new sequence of digits.

For example, suppose we have the following fractions:

1/3 = 0.3333...

1/7 = 0.142857142857...

1/9 = 0.1111...

1/11 = 0.090909...

...

We then construct a new number by taking the first digit after the decimal point of the first fraction (3), the second digit after the decimal point of the second fraction (4), the third digit after the decimal point of the third fraction (1), the fourth digit after the decimal point of the fourth fraction (0), and so on. Adding 1 (mod 10) to each digit, we get:

4 5 2 1 0 ...

Now, we claim that this number does not correspond to any fraction between 0 and 1. To see why, suppose that it does correspond to some fraction, say 0.a1a2a3..., where each ai is a digit between 0 and 9. Then, we can construct a new number by taking the first digit after the decimal point of the first fraction (which is different from a1), the second digit after the decimal point of the second fraction (which is different from a2), the third digit after the decimal point of the third fraction (which is different from a3), and so on. Adding 1 (mod 10) to each digit, we get a new sequence of digits that is different from the original number we constructed. Therefore, the original number cannot correspond to any fraction between 0 and 1.

This contradicts our assumption that there is no one-to-one correspondence between the set of all fractions between 0 and 1 and the set of all real numbers between 0 and infinity. Therefore, there must be a one-to-one correspondence between the two sets.