Order Properties of R - Density Property

I'ts funny how sometimes something small and innocuous turns out to be a bit of a problem. On Saturday I was looking at Exercise 1.6(b) on page 11 of AA1. The question asks:

Given positive real numbers a and b such that a<b, describe how you can find a rational number x and an irrational number y such that a<x<b and a<y<b.

The method is descibed in the text. Suppose that a=0.12333... and b=0.12345..., then the two decimals differ at the fourth digit after the decimal point. If we truncate b after this digit, then we obtain the rational number x=0.1234 which satisfies a<x<b. If we then add a sufficiently small non-recurring tail such as 010010001... to x then we obtain an irrational number y=0.1234010010001... which has a<y<b. Note that y is irrational because it is a non-recurring decimal (note also that the pattern of 010010001... is one zero 1, two zero's 1, three zero's 1 etc.).

What had me foxed was that in the solution it said:

We can arrange that a does not end in recurring 9s and b does not terminate.

I thought that this was a very odd statement and it took me a while to understand why it had been included in the solution. However, consider, for example, a=4.010... and b=4.100... Here a and b differ at the first digit after the decimal point, so we would want x to be b truncated to 4.1. However, b is already a terminating or finite decimal, so in this case you would have x=b rather than x<b. To get around this we use the fact that b is also equal to the recurring decimal 4.0999... i.e. we can arrange that b does not terminate. Now a and b differ at the second digit after the decimal point, so x would be b truncated to 4.09. Here x<b as required and now we can add a sufficiently small non-recurring tail to x to obtain y.

Consider another situation where a=3.999... and b=4.0123.. Here a and b differ at the first digit before the decimal point, so x would be b truncated to 4. The problem now is that 3.999.. i.e. 3.9 recurring is equal to 4 so now x=a, rather than x>a. The solution is that we can arrange that a does not end in recurring 9s. So if a=4.00... then x=4.01 and everything is ok again.