For each of the following, either prove the identity or provide a counterexample to show the identity is not always true.
- For all real
,
.
- For all real
, there exists a real
such that
.
- For every real
, there exists a real
such that
.
- There exists a real
such that
.
- False.
Counterexample: Let, then
, and
. Hence, we have
, but
; thus, the statement is false.
- False.
Counterexample: Let, then
, but
.
- True.
Proof. Let, then
for all
, and
for all
- True.
Proof. Let, then