inequalities for real numbers (original) (raw)

Suppose a is a real number.

2. 1. If a>0 then a is a positive number.
3. 1. If a≤0 then a is a non-positive number.
4. 1. If a≥0 then a is a non-negative number.

The first two inequalities are also called strict inequalities.
The second two inequalities are also called loose inequalities.

Properties

Suppose a and b are real numbers.

1. 1. If a>b, then -a<-b. If a<b, then -a>-b.
2. 1. If a≥b, then -a≤-b. If a≤b, then -a≥-b.

Proof.

If 0<a, then adding -a on both sides of the inequality gives -a=-a+0<-a+a=0. This process can also be reversed. ∎

Lemma 2.

For any a∈R, either a=0 or 0<a2.

Proof.

Suppose a≠0, then by trichotomy, we have either 0<a or a<0, but not both. If 0<a, then 0=0⋅a<a⋅a=a2. On the other hand, if -(-a)=a<0, then 0<-a by the previous lemma. Then repeating the previous , 0=0⋅(-a)<(-a)⁢(-a)=a2. ∎

Corollary 2.

For any a∈R, 0<1+a2.

Corollary 3.

There is no real solution for x in the equation 1+x2=0.

Inequality for a converging sequence

Suppose a0,a1,… is a sequence of real numbers converging to a real number a.

1. 1. If ai<b or ai≤bfor some real number b for each i, then a≤b.
2. 1. If ai>b or ai≥bfor some real number b for each i, then a≥b.