About rational number .?

Question

Given any rational number a and for any positive rational number ε , we can always find another rational number b such that │a-b│< ε .

Answer 1

Remember that the absolute value is the undirected distance between two values on the number line. The conjecture you've offered is that for each pair of rational a and ε with ε > 0, there exists rational b within ε of a.

│a - b│< ε implies ±(a - b) < ε

For a – b < 0,

│a - b│ = -(a – b)
│a - b│ = b – a

Thus,

b – a < ε

implies

b < ε + a

b is rational by the closure property.

For a – b ≥ 0,

│a - b│ = a – b

Thus,

a – b < ε

implies

a – ε < b

again, b is rational by the closure property.

Answer 2

Of course you can

Let b = a+ε/2