Decide which of that following are true statements Provide a

Decide which of that following are true statements. Provide a short justification for those that are valid and a counterexample for those that are not: (a) two real numbers satisfy a < b if and only if a < b+e for ever e>0 (b) two real numbers satisfy a < b if a < b+e for every e>0

Solution

Two real numbers satisfy a < b if and only if a < b + e for every e > 0

This statement is True.

Justification:

a < b implies that a < b + e for every e > 0. This is trivial to prove.

a < b implies that, a + e < b + e for every e > 0

Now, a < a + e for every e > 0

Therefore, we can write, a < a + e < b + e for every e > 0

In other words, a < b + e for every e > 0.

Now to prove the reverse statement.

Let, a < b + e for every e > 0

If we take limit_{e 0+} on both sides (i.e. e tends to 0 from the positive side), we get

a < b + 0 (as limit_{e 0+} e = 0)

a < b

(Proved)