Let x and y be two real numbers. Then any number between them may be written in the form ax+by, where a+b=1, a > 0, b > 0.

Now let x = 0.(9) and y = 1. According to you, 0.(9) is between these two numbers, such that ax+by = 0.(9) = x.

Rearranging this equation, we get:

ax + (1-a)y = x [substitute (1-a) for b]

ax + y - ay = x [expand (1-a)y]

y - ay = x - ax [subtract ax]

y(1-a) = x(1-a) [factorise]

y = x [divide by (1-a)]

But x = 0.(9) and y = 1. Therefore:

1 = 0.(9)

So you do agree that 0.(9) = 1. Glad you've finally seen the light.