> [!NOTE] **Theorem (Least upper bound property)** > Any non-empty [[Bound of Set of Reals|set of reals that is bounded above]] has a *least upper bound*. >*Proof*. ... > [!NOTE] Corollary (Greatest upper bound property) Any non-empty subset of $\mathbb{R}$ that is *bonded below* has a *greatest lower bound.* > *Proof*. Suppose that $S$ is a non-empty that is bounded below by $m$. Then $-S$ is a non-empty set that is bounded above by $-m$, so it has a least upper bound $-l$ by LUBA. > > Now $l$ is a lower bound for $S.$ Also if $-m$ is another upper bound for $-S$, we must have $-l \leq -m$, which shows that $l \geq m$ so that $-l$ is indeed the *greatest lower bound* for $S$. # Properties > [!NOTE] Theorem (Equivalence to Bolzano-Weirestrass) > - [[Bolzano-Weierstrass Theorem (Sequential Compactness of The Reals)]]; **Remark** - Suppose that we want to find a solution $x$ of a problem $P$. - Perhaps we can approximate $P$ by a sequence of problems $P_{n}$, which are easier to solve. - We can find solutions $x_{n}$of these approximate problems. - We would then like to show that $x_{n} → x$ and that $x$ is a solution of $P$. - Compactness results like the Bolzano–Weierstrass Theorem mean that if we can show that $x_{n}$ is bounded then we can at least find a subsequence $x_{n_{j}}$ that converges to some $x$; we then hope to be able to show that $x$ solves our initial problem $P$. # Applications - [[Archimedean Property of Real Numbers]]. - [[Monotone Bounded Real Sequence is Convergent]]; - [[Floor and Ceiling Functions]] & [[Floor and Ceiling Functions]] are well-defined - [[There is a unique real number that is the square root of 2]]; - [[Existence & uniqueness nth root of positive reals]] & [[Intermediate Value Theorem (IVT)]]. - [[Well-Ordering Principle]].