Real Analysis: Real Numbers and Completeness

Explain why a given interval will always contain both rational and irrational numbers for any delta greater than zero.

Explain why a given interval will always contain both rational and irrational numbers for any delta greater than zero.

Determine the infimum and supremum of the natural numbers.

Determine the infimum and supremum of the real numbers.

Determine the supremum and infimum of the set $\left\{ \frac{1}{n} : n \in \mathbb{N} \right\}$.

Determine the supremum and infimum of the set of all rational numbers whose square is less than two.

Suppose S is a non-empty subset of the real numbers that is bounded above by M. Then S has a least upper bound, meaning the supremum exists.

Provide an example of a set in the rational numbers that does not have a least upper bound.

Show that the set of rational numbers $Q$ does not have the least upper bound property by considering the set $A = \{ p \in \mathbb{Q} : p^2 < 2 \}$ and demonstrating that the supremum of this set is $\sqrt{2}$, which is not a rational number.

For the set of reciprocals of all natural numbers, find the greatest lower bound (infimum) and the least upper bound (supremum).

Given the set composed of terms rac{m}{m+n} for $m, n$ in natural numbers, find the greatest lower bound (infimum) and the least upper bound (supremum).

Prove that if $S$ is an upper bound for a set $A$, then $S$ is the least upper bound (supremum) of $A$ if and only if for all $epsilon > 0$, there exists an a  in A such that $S - epsilon < a$.

Take a subset of the rational numbers Q such that $x^2 < 2$ for all x in Q. Show that this set does not have a supremum in Q.

Prove that if $a$ and $b$ are real numbers and $a$ is positive, then there exists a natural number $n$ such that $n \times a > b$.

Prove that for any real number $x$, there exists a natural number greater than $x$, meaning the natural numbers are unbounded above.

Prove that the supremum of the set {$\displaystyle \frac{n}{n+1}$ | $n \in \mathbb{N}$} is 1.