- #36
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OK, so you know that f is integrable. So you know that [itex]sup\{s_p\}=inf\{S_p\}[/itex].
For -f, we define
[tex]m_i^\prime=inf\{-f(x)~\vert~x_{i-1}<x<x_i\}~\text{and}~M_i^\prime=sup\{-f(x)~\vert~x_{i-1}<x<x_i\}[/tex]
and
[tex]s_p^\prime=\sum m_i\Delta x_i~\text{and}~S_p^\prime=\sum M_i^\prime \Delta x_i[/tex]
You need to prove that [itex]\sup\{s_p^\prime\}=inf\{S_p^\prime\}[/itex] using the hypothesis [itex]sup\{s_p\}=inf\{S_p\}[/itex].
So, do you know a relation between [itex]m_i^\prime,~M^\prime_i,s_p^\prime,S_p^\prime[/itex] and [itex]m_i,~M_i,~s_p,~S_p[/itex]??
For -f, we define
[tex]m_i^\prime=inf\{-f(x)~\vert~x_{i-1}<x<x_i\}~\text{and}~M_i^\prime=sup\{-f(x)~\vert~x_{i-1}<x<x_i\}[/tex]
and
[tex]s_p^\prime=\sum m_i\Delta x_i~\text{and}~S_p^\prime=\sum M_i^\prime \Delta x_i[/tex]
You need to prove that [itex]\sup\{s_p^\prime\}=inf\{S_p^\prime\}[/itex] using the hypothesis [itex]sup\{s_p\}=inf\{S_p\}[/itex].
So, do you know a relation between [itex]m_i^\prime,~M^\prime_i,s_p^\prime,S_p^\prime[/itex] and [itex]m_i,~M_i,~s_p,~S_p[/itex]??