Prove or disprove if $int_{a}^{b} f$ exists and equals 0












5















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







share|cite|improve this question




















  • 1




    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.
    – robjohn♦
    Jan 4 at 15:43






  • 1




    @robjohn i made the necessary changes
    – Ashish K
    Jan 4 at 15:45
















5















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







share|cite|improve this question




















  • 1




    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.
    – robjohn♦
    Jan 4 at 15:43






  • 1




    @robjohn i made the necessary changes
    – Ashish K
    Jan 4 at 15:45














5












5








5








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







share|cite|improve this question
















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 4 at 15:47









Math_QED

7,31331450




7,31331450










asked Jan 4 at 15:36









Ashish KAshish K

815613




815613








  • 1




    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.
    – robjohn♦
    Jan 4 at 15:43






  • 1




    @robjohn i made the necessary changes
    – Ashish K
    Jan 4 at 15:45














  • 1




    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.
    – robjohn♦
    Jan 4 at 15:43






  • 1




    @robjohn i made the necessary changes
    – Ashish K
    Jan 4 at 15:45








1




1




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.
– robjohn♦
Jan 4 at 15:43




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.
– robjohn♦
Jan 4 at 15:43




1




1




@robjohn i made the necessary changes
– Ashish K
Jan 4 at 15:45




@robjohn i made the necessary changes
– Ashish K
Jan 4 at 15:45










1 Answer
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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



$$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$






share|cite|improve this answer





















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    1 Answer
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    active

    oldest

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    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2














    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



    $$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$






    share|cite|improve this answer


























      2














      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



      $$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$






      share|cite|improve this answer
























        2












        2








        2






        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



        $$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$






        share|cite|improve this answer












        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



        $$lim_{n to infty} (U(f,P_n)-L(f,P_n)) = 0$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 4 at 15:41









        Math_QEDMath_QED

        7,31331450




        7,31331450






























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