Suppose that x is real number. Prove that x+1/x =2 if and only
if x=1.
Prove that there does not exist a smallest positive real number.
Is the result still true if we replace ”real number” with
”integer”?
Suppose that x is a real number. Use either proof by
contrapositive or proof by contradiction to show that x3 + 5x = 0
implies that x = 0.
Prove or disprove the statements: (a) If x is a real number such
that |x + 2| + |x| ≤ 1, then x 2 + 2x − 1 ≤ 2.
(b) If x is a real number such that |x + 2| + |x| ≤ 2, then x 2
+ 2x − 1 ≤ 2.
(c) If x is a real number such that |x + 2| + |x| ≤ 3, then x 2
+ 2x − 1 ≤ 2....
Real Analysis: Prove a subset of the Reals is compact if and only
if it is closed and bounded. In other words, the set of reals
satisfies the Heine-Borel property.