epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Special Kinds of Theorems 41
and
a < b.
So, the statement "If ii, then i" is true.
Part 3. If i, then iii; that is:
By hypothesis:
If the number b is larger than the number a,
then their average is smaller than b.
a < b.
Because we want to obtain a + fc,we can add either a or ft to both sides of the
inequality. Because the conclusion we want to reach deals with b, we could
try adding b. Thus, we obtain:
a-^b
<2b.
Dividing by 2 yields
a + b .
This proves that the conclusion holds true. So, the statement "If i, then iii"
is true.
Part 4. If iii, then i; that is:
By hypothesis:
If the average of a and b, —-—, is smaller than b,
then b is larger than a.
a-^b
^
Then,
a + b <2b
and
a < b.
So, the statement "If iii, then i" is true.
Because statements ii and iii are both equivalent to statement i, they are
equivalent to each other. Thus, the proof is now complete. •