Answer:
It has been proven from the explanation.
Step-by-step explanation:
If we assume that the real number x is greater than 0. Now, If on the contrary, a is less than 0, we can make the argument that;
Let us make n to be a natural number and m = m(n) to be the largest positive integer in such a way that;
m/n < x
Thus, (m+1)/n ≥ x and finally;
|x - m/n| < 1/n
If r_n = m/n. It will be easy to show from the definition of limits that the sequence (r_n) has limit x.
From earlier where we assumed that x > 0, the numbers that will be obtained by truncating the decimal expansion of "x" at the n-th place will be rational, and clearly have the limit "x". Although it implies we are now assuming every real number will have a decimal expansion.
