Prove that the space of all real numbers is complete

Given any set of 15 integers, must there be two that have the sameremainder when divided by 12? Prove using the pigeonhole principle. Eitherdescribe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes, or describe a function by giving its domain, co-domain, and how elements of the domain are related to elements of the co-domain.

prove that or give a counter example , every set of negative real numbers has a supremum.

A number r ∈ R  is called algebraic provided for some n ∈ N there exists numbers  a0, a1, … , an ∈ Z such that r satisfies p( r ) = 0 where p ( x ) = a0 + a1x + a2x2 + ⋯ + anxn.   A number r ∈ R is called transcendental provided it is NOT algebraic prove the set of ALL algebraic numbers is countable and the set of ALL transcendental numbers is uncountable.