#law

1. (Commutativity of Addition)
2. (Associativity of Addition)
3. (Additive Identity of Zero)
4. and this y is denoted by (Existence of Additive Inverse)

#law

1. (Commutativity of Multiplication)
2. (Associativity of Multiplication)
3. (Existence of Multiplicative Identity)
4. this y is denoted by (Existence of Multiplicative Inverse of Reciprocal)

For exactly one of is always true.

#BasicRealAnalysis/Remark

#Proof/Contradiction
Assume
by Law of Trichotomy

Now assume that

Similarly is we assume that

#BasicRealAnalysis/Proposition

1. For , we have . The point is known as the midpoint between x and y. #Proof
2. If

#Proof/Contradiction

3. For , we have and , assuming the
   existence of . More generally, if , then .

#Proof/Induction

Therefore by induction
#Proof/Contradiction
To prove

Post Mail proof

4. For , we have
   #AMGM Proofs/AM-GM
   

Further AM-GM Proof