Prove or disprove if $int_{a}^{b} f$ exists and equals 0
$begingroup$
Prove or disprove: Suppose $f$ is bounded on the interval $[a,b]$ and
for any $ninmathbb{N}$ there exist partitions $P_{n}$ and $Q_{n}$ such that
$U(P_{n},f) le frac{1}{n}$ and $L(Q_{n},f) ge -frac{1}{n}$. Then
$int_{a}^{b} f$ exists and equals $0$.
Here's my attempt:
$$-frac{1}{n} overset{(1)}{leq} L(Q_{n},f) overset{(2)}{leq} underline{int_{a}^{b} f} overset{(3)}{leq} overline{int_{a}^{b} f} overset{(4)}{leq} U(P_{n},f) overset{(5)}{leq} frac{1}{n}$$
The inequalities $(1),(5)$ are given and $(2),(3),(4)$ hold by definition of lower integral and upper integral. It follows by squeeze theorem that $underline{int_{a}^{b} f} = overline{int_{a}^{b} f} = 0$.
Is this proof okay?
real-analysis proof-verification riemann-integration
asked Jan 4 at 15:36
Ashish KAshish K
923614
$endgroup$
add a comment |
$begingroup$
Prove or disprove: Suppose $f$ is bounded on the interval $[a,b]$ and
for any $ninmathbb{N}$ there exist partitions $P_{n}$ and $Q_{n}$ such that
$U(P_{n},f) le frac{1}{n}$ and $L(Q_{n},f) ge -frac{1}{n}$. Then
$int_{a}^{b} f$ exists and equals $0$.
Here's my attempt:
$$-frac{1}{n} overset{(1)}{leq} L(Q_{n},f) overset{(2)}{leq} underline{int_{a}^{b} f} overset{(3)}{leq} overline{int_{a}^{b} f} overset{(4)}{leq} U(P_{n},f) overset{(5)}{leq} frac{1}{n}$$
The inequalities $(1),(5)$ are given and $(2),(3),(4)$ hold by definition of lower integral and upper integral. It follows by squeeze theorem that $underline{int_{a}^{b} f} = overline{int_{a}^{b} f} = 0$.
Is this proof okay?
real-analysis proof-verification riemann-integration
asked Jan 4 at 15:36
Ashish KAshish K
923614
$endgroup$
1
$begingroup$
After a while, I realized that $P_n$ and $Q_n$ are partitions. However, it would be nice if that were mentioned in the question. Otherwise, your proof looks good.
$endgroup$
– robjohn♦
Jan 4 at 15:43
1
$begingroup$
@robjohn i made the necessary changes
$endgroup$
– Ashish K
Jan 4 at 15:45
add a comment |
$begingroup$
Prove or disprove: Suppose $f$ is bounded on the interval $[a,b]$ and
for any $ninmathbb{N}$ there exist partitions $P_{n}$ and $Q_{n}$ such that
$U(P_{n},f) le frac{1}{n}$ and $L(Q_{n},f) ge -frac{1}{n}$. Then
$int_{a}^{b} f$ exists and equals $0$.
Here's my attempt:
$$-frac{1}{n} overset{(1)}{leq} L(Q_{n},f) overset{(2)}{leq} underline{int_{a}^{b} f} overset{(3)}{leq} overline{int_{a}^{b} f} overset{(4)}{leq} U(P_{n},f) overset{(5)}{leq} frac{1}{n}$$
The inequalities $(1),(5)$ are given and $(2),(3),(4)$ hold by definition of lower integral and upper integral. It follows by squeeze theorem that $underline{int_{a}^{b} f} = overline{int_{a}^{b} f} = 0$.
Is this proof okay?
real-analysis proof-verification riemann-integration
asked Jan 4 at 15:36
Ashish KAshish K
923614
$endgroup$
Prove or disprove: Suppose $f$ is bounded on the interval $[a,b]$ and
for any $ninmathbb{N}$ there exist partitions $P_{n}$ and $Q_{n}$ such that
$U(P_{n},f) le frac{1}{n}$ and $L(Q_{n},f) ge -frac{1}{n}$. Then
$int_{a}^{b} f$ exists and equals $0$.
Here's my attempt:
$$-frac{1}{n} overset{(1)}{leq} L(Q_{n},f) overset{(2)}{leq} underline{int_{a}^{b} f} overset{(3)}{leq} overline{int_{a}^{b} f} overset{(4)}{leq} U(P_{n},f) overset{(5)}{leq} frac{1}{n}$$
The inequalities $(1),(5)$ are given and $(2),(3),(4)$ hold by definition of lower integral and upper integral. It follows by squeeze theorem that $underline{int_{a}^{b} f} = overline{int_{a}^{b} f} = 0$.
Is this proof okay?
real-analysis proof-verification riemann-integration
real-analysis proof-verification riemann-integration
asked Jan 4 at 15:36
Ashish KAshish K
923614
asked Jan 4 at 15:36
Ashish KAshish K
923614
edited Jan 4 at 15:47
user370967
asked Jan 4 at 15:36
Ashish KAshish K
923614
asked Jan 4 at 15:36
Ashish KAshish K
923614
asked Jan 4 at 15:36
Ashish KAshish K
923614
923614
1
$begingroup$
After a while, I realized that $P_n$ and $Q_n$ are partitions. However, it would be nice if that were mentioned in the question. Otherwise, your proof looks good.
$endgroup$
– robjohn♦
Jan 4 at 15:43
1
$begingroup$
@robjohn i made the necessary changes
$endgroup$
– Ashish K
Jan 4 at 15:45
add a comment |
1
$begingroup$
After a while, I realized that $P_n$ and $Q_n$ are partitions. However, it would be nice if that were mentioned in the question. Otherwise, your proof looks good.
$endgroup$
– robjohn♦
Jan 4 at 15:43
1
$begingroup$
@robjohn i made the necessary changes
$endgroup$
– Ashish K
Jan 4 at 15:45
1
1
$begingroup$
After a while, I realized that $P_n$ and $Q_n$ are partitions. However, it would be nice if that were mentioned in the question. Otherwise, your proof looks good.
$endgroup$
– robjohn♦
Jan 4 at 15:43
$begingroup$
After a while, I realized that $P_n$ and $Q_n$ are partitions. However, it would be nice if that were mentioned in the question. Otherwise, your proof looks good.
$endgroup$
– robjohn♦
Jan 4 at 15:43
1
1
$begingroup$
@robjohn i made the necessary changes
$endgroup$
– Ashish K
Jan 4 at 15:45
$begingroup$
@robjohn i made the necessary changes
$endgroup$
– Ashish K
Jan 4 at 15:45
add a comment |
1 Answer
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$begingroup$
Your proof is completely correct. Well done!
Note also that the following related statement is true:
A bounded function $f: [a,b] to mathbb{R}$ is Riemann-integrable if and only if there is a sequence of partitions $(P_n)$ such that
$$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$
$endgroup$
add a comment |
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1 Answer
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active
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$begingroup$
Your proof is completely correct. Well done!
Note also that the following related statement is true:
A bounded function $f: [a,b] to mathbb{R}$ is Riemann-integrable if and only if there is a sequence of partitions $(P_n)$ such that
$$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$
$endgroup$
add a comment |
$begingroup$
Your proof is completely correct. Well done!
Note also that the following related statement is true:
A bounded function $f: [a,b] to mathbb{R}$ is Riemann-integrable if and only if there is a sequence of partitions $(P_n)$ such that
$$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$
$endgroup$
add a comment |
$begingroup$
Your proof is completely correct. Well done!
Note also that the following related statement is true:
A bounded function $f: [a,b] to mathbb{R}$ is Riemann-integrable if and only if there is a sequence of partitions $(P_n)$ such that
$$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$
$endgroup$
Your proof is completely correct. Well done!
Note also that the following related statement is true:
A bounded function $f: [a,b] to mathbb{R}$ is Riemann-integrable if and only if there is a sequence of partitions $(P_n)$ such that
$$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$
answered Jan 4 at 15:41
user370967
add a comment |
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$begingroup$
After a while, I realized that $P_n$ and $Q_n$ are partitions. However, it would be nice if that were mentioned in the question. Otherwise, your proof looks good.
$endgroup$
– robjohn♦
Jan 4 at 15:43
1
$begingroup$
@robjohn i made the necessary changes
$endgroup$
– Ashish K
Jan 4 at 15:45