prove $L(f)leq U(f)$
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How exactly would I go about proving the following statement?
Given $f:[a,b]tomathbb{R}$ show that $$L(f)leq U(f)$$ where $$L(f)=sup_{Pinmathscr{P}}L(f,P) text{ and } U(f)=inf_{Pinmathscr{P}}U(f,P)$$ where $P$ is any partition of $[a,b]$ and $mathscr{P}$ is the set of all partitions of $[a,b]$.
Intuitively this makes sense because for any partition $P$ made up of the intervals $I_{1},...,I_{n}$ we know that $$m_{k}=inf_{xin I_{k}}f(x)leq M_{k}=sup_{xin I_{k}}f(x)$$
Would I be along the right lines to consider $P$ being the partition used for $L(f)$ and $Q$ being the partition used for $U(f)$ and then let $R$ be a partition defined as $R=Pcup Q$ so that $R$ is a refinement of both $P$ and $Q$?
Thanks!
integration analysis riemann-integration partitions-for-integration
$endgroup$
add a comment |
$begingroup$
How exactly would I go about proving the following statement?
Given $f:[a,b]tomathbb{R}$ show that $$L(f)leq U(f)$$ where $$L(f)=sup_{Pinmathscr{P}}L(f,P) text{ and } U(f)=inf_{Pinmathscr{P}}U(f,P)$$ where $P$ is any partition of $[a,b]$ and $mathscr{P}$ is the set of all partitions of $[a,b]$.
Intuitively this makes sense because for any partition $P$ made up of the intervals $I_{1},...,I_{n}$ we know that $$m_{k}=inf_{xin I_{k}}f(x)leq M_{k}=sup_{xin I_{k}}f(x)$$
Would I be along the right lines to consider $P$ being the partition used for $L(f)$ and $Q$ being the partition used for $U(f)$ and then let $R$ be a partition defined as $R=Pcup Q$ so that $R$ is a refinement of both $P$ and $Q$?
Thanks!
integration analysis riemann-integration partitions-for-integration
$endgroup$
$begingroup$
Can you say anything about the relative values of any two given lower and upper partitions?
$endgroup$
– AnotherJohnDoe
Dec 18 '18 at 9:29
$begingroup$
How do you guarantee the existence of $P,Q$?
$endgroup$
– Shubham Johri
Dec 18 '18 at 9:43
add a comment |
$begingroup$
How exactly would I go about proving the following statement?
Given $f:[a,b]tomathbb{R}$ show that $$L(f)leq U(f)$$ where $$L(f)=sup_{Pinmathscr{P}}L(f,P) text{ and } U(f)=inf_{Pinmathscr{P}}U(f,P)$$ where $P$ is any partition of $[a,b]$ and $mathscr{P}$ is the set of all partitions of $[a,b]$.
Intuitively this makes sense because for any partition $P$ made up of the intervals $I_{1},...,I_{n}$ we know that $$m_{k}=inf_{xin I_{k}}f(x)leq M_{k}=sup_{xin I_{k}}f(x)$$
Would I be along the right lines to consider $P$ being the partition used for $L(f)$ and $Q$ being the partition used for $U(f)$ and then let $R$ be a partition defined as $R=Pcup Q$ so that $R$ is a refinement of both $P$ and $Q$?
Thanks!
integration analysis riemann-integration partitions-for-integration
$endgroup$
How exactly would I go about proving the following statement?
Given $f:[a,b]tomathbb{R}$ show that $$L(f)leq U(f)$$ where $$L(f)=sup_{Pinmathscr{P}}L(f,P) text{ and } U(f)=inf_{Pinmathscr{P}}U(f,P)$$ where $P$ is any partition of $[a,b]$ and $mathscr{P}$ is the set of all partitions of $[a,b]$.
Intuitively this makes sense because for any partition $P$ made up of the intervals $I_{1},...,I_{n}$ we know that $$m_{k}=inf_{xin I_{k}}f(x)leq M_{k}=sup_{xin I_{k}}f(x)$$
Would I be along the right lines to consider $P$ being the partition used for $L(f)$ and $Q$ being the partition used for $U(f)$ and then let $R$ be a partition defined as $R=Pcup Q$ so that $R$ is a refinement of both $P$ and $Q$?
Thanks!
integration analysis riemann-integration partitions-for-integration
integration analysis riemann-integration partitions-for-integration
edited Dec 18 '18 at 9:38
Christian Blatter
173k8113326
173k8113326
asked Dec 18 '18 at 9:26
BigWigBigWig
11310
11310
$begingroup$
Can you say anything about the relative values of any two given lower and upper partitions?
$endgroup$
– AnotherJohnDoe
Dec 18 '18 at 9:29
$begingroup$
How do you guarantee the existence of $P,Q$?
$endgroup$
– Shubham Johri
Dec 18 '18 at 9:43
add a comment |
$begingroup$
Can you say anything about the relative values of any two given lower and upper partitions?
$endgroup$
– AnotherJohnDoe
Dec 18 '18 at 9:29
$begingroup$
How do you guarantee the existence of $P,Q$?
$endgroup$
– Shubham Johri
Dec 18 '18 at 9:43
$begingroup$
Can you say anything about the relative values of any two given lower and upper partitions?
$endgroup$
– AnotherJohnDoe
Dec 18 '18 at 9:29
$begingroup$
Can you say anything about the relative values of any two given lower and upper partitions?
$endgroup$
– AnotherJohnDoe
Dec 18 '18 at 9:29
$begingroup$
How do you guarantee the existence of $P,Q$?
$endgroup$
– Shubham Johri
Dec 18 '18 at 9:43
$begingroup$
How do you guarantee the existence of $P,Q$?
$endgroup$
– Shubham Johri
Dec 18 '18 at 9:43
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Say $displaystyle L(f)>U(f)=inf_{Pinscr P}U(f,P)$
$L(f)$ is not a lower bound for ${U(f,P):Pinmathscr P}thereforeexists P_1inmathscr P$ such that $U(f)le U(f,P_1)<L(f)$. $U(f,P_1)$ is not an upper bound for ${L(f,P):pinmathscr P}therefore exists P_2inmathscr P$ such that $U(f,P_1)<L(f,P_2)le L(f)$.
This is a contradiction, since $U(f,P_1)ge L(f,P_2) forall P_1,P_2inmathscr P$.
$endgroup$
add a comment |
$begingroup$
If $P,Q in mathscr{P}$, then we have
$L(F,P) le U(f,Q)$.
This gives
$L(f) le U(f,Q)$ for all $Q in mathscr{P}$,
hence $L(f) le U(f).$
$endgroup$
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
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oldest
votes
$begingroup$
Say $displaystyle L(f)>U(f)=inf_{Pinscr P}U(f,P)$
$L(f)$ is not a lower bound for ${U(f,P):Pinmathscr P}thereforeexists P_1inmathscr P$ such that $U(f)le U(f,P_1)<L(f)$. $U(f,P_1)$ is not an upper bound for ${L(f,P):pinmathscr P}therefore exists P_2inmathscr P$ such that $U(f,P_1)<L(f,P_2)le L(f)$.
This is a contradiction, since $U(f,P_1)ge L(f,P_2) forall P_1,P_2inmathscr P$.
$endgroup$
add a comment |
$begingroup$
Say $displaystyle L(f)>U(f)=inf_{Pinscr P}U(f,P)$
$L(f)$ is not a lower bound for ${U(f,P):Pinmathscr P}thereforeexists P_1inmathscr P$ such that $U(f)le U(f,P_1)<L(f)$. $U(f,P_1)$ is not an upper bound for ${L(f,P):pinmathscr P}therefore exists P_2inmathscr P$ such that $U(f,P_1)<L(f,P_2)le L(f)$.
This is a contradiction, since $U(f,P_1)ge L(f,P_2) forall P_1,P_2inmathscr P$.
$endgroup$
add a comment |
$begingroup$
Say $displaystyle L(f)>U(f)=inf_{Pinscr P}U(f,P)$
$L(f)$ is not a lower bound for ${U(f,P):Pinmathscr P}thereforeexists P_1inmathscr P$ such that $U(f)le U(f,P_1)<L(f)$. $U(f,P_1)$ is not an upper bound for ${L(f,P):pinmathscr P}therefore exists P_2inmathscr P$ such that $U(f,P_1)<L(f,P_2)le L(f)$.
This is a contradiction, since $U(f,P_1)ge L(f,P_2) forall P_1,P_2inmathscr P$.
$endgroup$
Say $displaystyle L(f)>U(f)=inf_{Pinscr P}U(f,P)$
$L(f)$ is not a lower bound for ${U(f,P):Pinmathscr P}thereforeexists P_1inmathscr P$ such that $U(f)le U(f,P_1)<L(f)$. $U(f,P_1)$ is not an upper bound for ${L(f,P):pinmathscr P}therefore exists P_2inmathscr P$ such that $U(f,P_1)<L(f,P_2)le L(f)$.
This is a contradiction, since $U(f,P_1)ge L(f,P_2) forall P_1,P_2inmathscr P$.
answered Dec 18 '18 at 9:43
Shubham JohriShubham Johri
5,192717
5,192717
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add a comment |
$begingroup$
If $P,Q in mathscr{P}$, then we have
$L(F,P) le U(f,Q)$.
This gives
$L(f) le U(f,Q)$ for all $Q in mathscr{P}$,
hence $L(f) le U(f).$
$endgroup$
add a comment |
$begingroup$
If $P,Q in mathscr{P}$, then we have
$L(F,P) le U(f,Q)$.
This gives
$L(f) le U(f,Q)$ for all $Q in mathscr{P}$,
hence $L(f) le U(f).$
$endgroup$
add a comment |
$begingroup$
If $P,Q in mathscr{P}$, then we have
$L(F,P) le U(f,Q)$.
This gives
$L(f) le U(f,Q)$ for all $Q in mathscr{P}$,
hence $L(f) le U(f).$
$endgroup$
If $P,Q in mathscr{P}$, then we have
$L(F,P) le U(f,Q)$.
This gives
$L(f) le U(f,Q)$ for all $Q in mathscr{P}$,
hence $L(f) le U(f).$
answered Dec 18 '18 at 9:46
FredFred
46.7k1848
46.7k1848
add a comment |
add a comment |
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$begingroup$
Can you say anything about the relative values of any two given lower and upper partitions?
$endgroup$
– AnotherJohnDoe
Dec 18 '18 at 9:29
$begingroup$
How do you guarantee the existence of $P,Q$?
$endgroup$
– Shubham Johri
Dec 18 '18 at 9:43