What is a counterexample to show that the repeating decimals are closed under subtraction false?

Answer 1

In fact, the statement is true. Consequently, there is not a proper counterexample. The fallacy is in asserting that a terminating decimal is not a repeating decimal.

First, there is the trivial argument that any terminating decimal can be written with a repeating string of trailing zeros. But, Cantor or Dedekind (I can't remember which) proved that any terminating decimal can also be expressed as a repeating decimal.

For example, 2.35 can be written as 2.3499...

Or 150,000 as 149,999.99...

Thus, a terminating decimal becomes a recurring decimal. As a consequence, all real numbers can be expressed as infinite decimals. And that proves closure under addition.

Answer 2

False because they will always repeating