Use the Reflection Principle and analogous results about suprema to prove the following results. a) [APPROXIMATION PROPERTY

Question:

Use the Reflection Principle and analogous results about suprema to prove the following results.
a) [APPROXIMATION PROPERTY FOR INFIMA] Prove that if a set E ⊂ R has a finite infimum and ε > 0 is any positive number, then there is a point a ∈ E such that inf E + ε > a > inf E.
b) [COMPLETENESS PROPERTY FOR INFIMA] If E ⊂ R is nonempty and bounded below, then E has a (finite) infimum.