Hello amm345We need to establish first that .

Now one of the axioms of an ordered field is that ; and it is well established that (see here for instance.)

So

, since

We can now prove the first of the required statements by contradiction,as follows:

First consider the case where and assume that . Then:

using the result above.

But . Contradiction. Therefore .

Then prove it the other way around, and consider the case where , and assume that .

, again using the same result.

But . Contradiction. Therefore

We have now proved the implication in both directions. Hence

To prove the second of your two statements, establish first that .Hint:for some . Then use the distributive law on .

Then use a contradiction method as I did for the first part, to show that . Hence, using the result above that .

Finally, prove the result in the other direction to show that .

Grandad