HelmutKnaust at 14:34, 27 August 2014

2014-08-27T14:34:27Z

HelmutKnaust: Created page with "'''Problem 1.''' Exercise 1.3.2. '''Problem 2.''' Exercise 1.3.3(a)(b). '''Problem 3.''' Let $A$ be a non-empty set of real numbers that is bounded from above. Show: If $s$ ..."

2014-08-27T14:33:55Z

Created page with "'''Problem 1.''' Exercise 1.3.2. '''Problem 2.''' Exercise 1.3.3(a)(b). '''Problem 3.''' Let $A$ be a non-empty set of real numbers that is bounded from above. Show: If $s$ ..."

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'''Problem 1.''' Exercise 1.3.2.

'''Problem 2.''' Exercise 1.3.3(a)(b).

'''Problem 3.''' Let $A$ be a non-empty set of real numbers that is bounded from above. Show: If $s$ and $t$ both are suprema of $A$, then $s=t$. (Suprema are unique.)

'''Problem 4.''' A real number $m\in\mathbb{R}$ is called the maximum of the set $A\subseteq\mathbb{R}$, if $m\in A$ and $m\geq a$ for all $a\in A$.
# Show: If $m$ is the maximum of $A$, then $m$ is also the supremum of $A$.
# Let $A=\{x\in\mathbb{Q}\ |\ x^2\leq 5\}$. Show that A is bounded from above, but that $A$ has no maximum.

'''Problem 5.''' Show that the ''Nested Interval Property'' together with the ''Archimedean Principle'' implies the ''Axiom of Completeness''.

@@ Line 5: / Line 5: @@
 '''Problem 3.''' Let $A$ be a non-empty set of real numbers that is bounded from above. Show: If $s$ and $t$ both are suprema of $A$, then $s=t$. (Suprema are unique.)
-'''Problem 4.''' A real number $m\in\mathbb{R}$ is called the maximum of the set $A\subseteq\mathbb{R}$, if $m\in A$ and $m\geq a$ for all $a\in A$.
+'''Problem 4.''' A real number $m$ is called maximum of the set $A\subseteq\mathbb{R}$, if $m\in A$ and $m\geq a$ for all $a\in A$.
 # Show: If $m$ is the maximum of $A$, then $m$ is also the supremum of $A$.
 # Let $A=\{x\in\mathbb{Q}\ |\ x^2\leq 5\}$. Show that A is bounded from above, but that $A$ has no maximum.
 '''Problem 5.''' Show that the ''Nested Interval Property'' together with the ''Archimedean Principle'' implies the ''Axiom of Completeness''.

CRN 11982: HW 1 - Revision history

HelmutKnaust at 14:34, 27 August 2014

HelmutKnaust: Created page with "'''Problem 1.''' Exercise 1.3.2. '''Problem 2.''' Exercise 1.3.3(a)(b). '''Problem 3.''' Let $A$ be a non-empty set of real numbers that is bounded from above. Show: If $s$ ..."