Porve Between any two real numbers lies and algebraic number

Porve \"Between any two real numbers lies and algebraic number and also a transcendental number\".

Solution

Prove:

Every interval is uncountable and R has larger cardinality then N.

Every open interval (a,b) have the same cardinality as R

All open intervals have the same cardinality.

how to prove cardinality:

if we can prove that Real Numbers and the interval (0,) have the same cardinality then we can say between any 2 real numbers lies algebraic number and transcendental number.

Consider the function f(x)=e^x

The domain of this function is all real numbers.

The range of this function is from 0 to infinity.

Let e^a=e^b .

Then ln(e^a) = ln(e^b)

Then a ln(e) = b ln(e)

This means that    a = b

Hence, f is injective.

Let   c > 0

Then    e^ln(c) = c

Since c > 0, ln(c) is defined, so    f (ln(c)) = c

Therefore, f is surjective.

Then f is bijective.

Hence, R and (0,) have the same cardinality.

so we can say that Between any two real numbers lies and algebraic number and also a transcendental number