CRN 11378: HW 1 - Revision history

HelmutKnaust at 19:01, 9 October 2019

2019-10-09T19:01:36Z

HelmutKnaust at 05:12, 3 September 2019

2019-09-03T05:12:44Z

HelmutKnaust at 05:09, 3 September 2019

2019-09-03T05:09:02Z

HelmutKnaust at 05:07, 3 September 2019

2019-09-03T05:07:27Z

HelmutKnaust at 05:06, 3 September 2019

2019-09-03T05:06:27Z

HelmutKnaust: Created page with "'''Problem 1.''' If $\sup A < \sup B$, show that $B$ contains an element that is an upper bound of $A$. '''Problem 2.''' Let $A$ be a non-empty set of real numbers that is b..."

2019-09-03T05:06:13Z

Created page with "'''Problem 1.''' If $\sup A < \sup B$, show that $B$ contains an element that is an upper bound of $A$. '''Problem 2.''' Let $A$ be a non-empty set of real numbers that is b..."

New page

'''Problem 1.''' If $\sup A < \sup B$, show that $B$ contains an element that is an upper bound of $A$.

'''Problem 2.''' 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 3.''' Suppose every non-empty set that is bounded form above has a supremum. Show that then every non-empty set that is bounded form below has an infimum.

'''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''.

@@ Line 3: / Line 3: @@
 '''Problem 2.''' 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 $s=t$. (This shows that a set can have at most one supremum.)
-'''Problem 3.''' Suppose that every non-empty set that is bounded form above has a supremum. Show that then every non-empty set that is bounded form below has an infimum.
+'''Problem 3.''' Suppose that every non-empty set that is bounded from above has a supremum. Show that then every non-empty set that is bounded form below has an infimum.
 '''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 $a\in A$.

@@ Line 1: / Line 1: @@
 '''Problem 1.''' Let $A$ and $B$ be two non-empty sets that are bounded from above. Show: If $\sup A < \sup B$, then $B$ contains an element that is an upper bound of $A$.
-'''Problem 2.''' 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 2.''' 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 $s=t$. (This shows that a set can have at most one supremum.)
 '''Problem 3.''' Suppose that every non-empty set that is bounded form above has a supremum. Show that then every non-empty set that is bounded form below has an infimum.

@@ Line 1: / Line 1: @@
-'''Problem 1.''' Let $A$ and $B$ be two non-empty sets that are bounded from above. Show: If $\sup A < \sup B$, show that $B$ contains an element that is an upper bound of $A$.
+'''Problem 1.''' Let $A$ and $B$ be two non-empty sets that are bounded from above. Show: If $\sup A < \sup B$, then $B$ contains an element that is an upper bound of $A$.
 '''Problem 2.''' 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 3.''' Suppose every non-empty set that is bounded form above has a supremum. Show that then every non-empty set that is bounded form below has an infimum.
+'''Problem 3.''' Suppose that every non-empty set that is bounded form above has a supremum. Show that then every non-empty set that is bounded form below has an infimum.
 '''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 $a\in A$.

← Older revision		Revision as of 05:07, 3 September 2019
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−	'''Problem 1.''' If $\sup A < \sup B$, show that $B$ contains an element that is an upper bound of $A$.	+	'''Problem 1.''' Let $A$ and $B$ be two non-empty sets that are bounded from above. Show: If $\sup A < \sup B$, show that $B$ contains an element that is an upper bound of $A$.

	'''Problem 2.''' 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 2.''' 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.)

← Older revision		Revision as of 05:06, 3 September 2019
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	'''Problem 1.''' If $\sup A < \sup B$, show that $B$ contains an element that is an upper bound of $A$.		'''Problem 1.''' If $\sup A < \sup B$, show that $B$ contains an element that is an upper bound of $A$.
−

	'''Problem 2.''' 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 2.''' 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.)