Simple question: Writing numers into decimal

Question

I want to know if there are numbers in the interval $[\frac{28}{100},\frac{29}{100}]$

which have a $7$ in their decimal expansion.

I would say "yes", because we can write $0.28=0.2799999999...$

But this is the only number in this interval, right?

Answer 1

There are infinite numbers in your interval, and infinite of them have a $7$ in their decimal expansion. All the numbers that start with $0.28$ in decimal notation, followed by anything that contains a $7$, satisfy your requirement.

Examples: $0.287$, $0.2807$, $0.2817$, $0.2871$, $0.280000000000000007$

Notice that this is not limited to the rationals. For example

$0.28 + e/1000$

also satisfies the required condition.