prove $U(P_n,f)−L(P_n,f)$ = $ frac{f(1)−f(0)}{n}$
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Suppose $ f: [0,1]$ -> $R$ is an increasing function. For each $ n in N$ , consider the uniform partition $Pn := ( x_0, x_1,..., x_n )$ of $[0,1]$ where $x_i = i$ for $0≤i≤n$.
Now, I need to prove
a) $U(P_n,f)−L(P_n,f)$ = $ frac{f(1)−f(0)}{n}$
and
b) show $f in R [0, 1]$
I get some idea on part a: $m_i = frac {x_i}{n} $ and $M_i= {x_i}{n}$ , then I am not sure
but for part b, I have no idea (is it something like $epsilon > 0$ )?
real-analysis proof-verification
asked Nov 22 at 20:05
Math Avengers
709
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Suppose $ f: [0,1]$ -> $R$ is an increasing function. For each $ n in N$ , consider the uniform partition $Pn := ( x_0, x_1,..., x_n )$ of $[0,1]$ where $x_i = i$ for $0≤i≤n$.
Now, I need to prove
a) $U(P_n,f)−L(P_n,f)$ = $ frac{f(1)−f(0)}{n}$
and
b) show $f in R [0, 1]$
I get some idea on part a: $m_i = frac {x_i}{n} $ and $M_i= {x_i}{n}$ , then I am not sure
but for part b, I have no idea (is it something like $epsilon > 0$ )?
real-analysis proof-verification
asked Nov 22 at 20:05
Math Avengers
709
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose $ f: [0,1]$ -> $R$ is an increasing function. For each $ n in N$ , consider the uniform partition $Pn := ( x_0, x_1,..., x_n )$ of $[0,1]$ where $x_i = i$ for $0≤i≤n$.
Now, I need to prove
a) $U(P_n,f)−L(P_n,f)$ = $ frac{f(1)−f(0)}{n}$
and
b) show $f in R [0, 1]$
I get some idea on part a: $m_i = frac {x_i}{n} $ and $M_i= {x_i}{n}$ , then I am not sure
but for part b, I have no idea (is it something like $epsilon > 0$ )?
real-analysis proof-verification
asked Nov 22 at 20:05
Math Avengers
709
Suppose $ f: [0,1]$ -> $R$ is an increasing function. For each $ n in N$ , consider the uniform partition $Pn := ( x_0, x_1,..., x_n )$ of $[0,1]$ where $x_i = i$ for $0≤i≤n$.
Now, I need to prove
a) $U(P_n,f)−L(P_n,f)$ = $ frac{f(1)−f(0)}{n}$
and
b) show $f in R [0, 1]$
I get some idea on part a: $m_i = frac {x_i}{n} $ and $M_i= {x_i}{n}$ , then I am not sure
but for part b, I have no idea (is it something like $epsilon > 0$ )?
real-analysis proof-verification
real-analysis proof-verification
asked Nov 22 at 20:05
Math Avengers
709
asked Nov 22 at 20:05
Math Avengers
709
asked Nov 22 at 20:05
Math Avengers
709
asked Nov 22 at 20:05
Math Avengers
709
asked Nov 22 at 20:05
Math Avengers
709
709
add a comment |
add a comment |
1 Answer
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Abridged solution. If $f$ is increasing on $[a, b]$ then $suplimits_{x in [a, b]} f(x) = f(b)$ and $inflimits_{x in [a, b]} f(x) = f(a).$ Q.E.D.
Addendum. Part (b) follows instantaneously from part (a) if you know the definition of Riemann integral as the equality if the supremum of lower approximations and infimum of upper approximations.
answered Nov 22 at 20:08
Will M.
2,337313
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1 Answer
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1 Answer
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Abridged solution. If $f$ is increasing on $[a, b]$ then $suplimits_{x in [a, b]} f(x) = f(b)$ and $inflimits_{x in [a, b]} f(x) = f(a).$ Q.E.D.
Addendum. Part (b) follows instantaneously from part (a) if you know the definition of Riemann integral as the equality if the supremum of lower approximations and infimum of upper approximations.
answered Nov 22 at 20:08
Will M.
2,337313
add a comment |
up vote
0
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Abridged solution. If $f$ is increasing on $[a, b]$ then $suplimits_{x in [a, b]} f(x) = f(b)$ and $inflimits_{x in [a, b]} f(x) = f(a).$ Q.E.D.
Addendum. Part (b) follows instantaneously from part (a) if you know the definition of Riemann integral as the equality if the supremum of lower approximations and infimum of upper approximations.
answered Nov 22 at 20:08
Will M.
2,337313
add a comment |
up vote
0
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up vote
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Abridged solution. If $f$ is increasing on $[a, b]$ then $suplimits_{x in [a, b]} f(x) = f(b)$ and $inflimits_{x in [a, b]} f(x) = f(a).$ Q.E.D.
Addendum. Part (b) follows instantaneously from part (a) if you know the definition of Riemann integral as the equality if the supremum of lower approximations and infimum of upper approximations.
answered Nov 22 at 20:08
Will M.
2,337313
Abridged solution. If $f$ is increasing on $[a, b]$ then $suplimits_{x in [a, b]} f(x) = f(b)$ and $inflimits_{x in [a, b]} f(x) = f(a).$ Q.E.D.
Addendum. Part (b) follows instantaneously from part (a) if you know the definition of Riemann integral as the equality if the supremum of lower approximations and infimum of upper approximations.
answered Nov 22 at 20:08
Will M.
2,337313
edited Nov 22 at 20:16
answered Nov 22 at 20:08
Will M.
2,337313
answered Nov 22 at 20:08
Will M.
2,337313
answered Nov 22 at 20:08
Will M.
2,337313
2,337313
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