Show that π(βπ, π) = βπΏ(π, π) and πΏ(βπ, π) = βπ(π, π)
let p is partition on [a,b]
"p=\\{a=x_0, x_1, x_2...x_{i-1}, x_i,...x_n=b \\}"
is any partition on [a,b]
By definition lower and upper Riemman sum
"L(f,p)=\\displaystyle\\sum_{i=1}^nm_i\\Delta x_i"
"U(f,p)=\\displaystyle\\sum_{i=1}^nm_i \\Delta x_i"
where "m_i=int \\{f(x)x\\in[x_{i-1},x_i]\\}"
"m_i=sup\\{f(x):x\\in[x_{i-1},x_i]\\}"
"intrimum(-f(x))=-supremum \\space f(x)"
"m_i=int \\{-f(x)x\\in[x_{i-1},x_i]\\}"
"=-sup\\{f(x):x\\in[x_{i-1},x_i]\\}"
"=-m_i"
"L(-f,p)=\\displaystyle\\sum_{i=1}^nm_i\\Delta x_i"
"=-\\displaystyle\\sum_{i=1}^nm_i\\Delta x_i"
"\ud835\udc3f(\u2212\ud835\udc53, \ud835\udc5d) = \u2212\ud835\udc48(\ud835\udc53, \ud835\udc5d).......proved"
now,
"m_i=sup\\{-f(x):x\\in[x_{i-1},x_i]\\}"
"=-int \\{f(x)x\\in[x_{i-1},x_i]\\}\\\\=-m_i"
"U(-f,p)=\\displaystyle\\sum_{i=1}^nm_i\\Delta x_i"
"=-\\displaystyle\\sum_{i=1}^nm_i\\Delta x_i"
"\ud835\udc48(\u2212\ud835\udc53, \ud835\udc5d) = \u2212\ud835\udc3f(\ud835\udc53, \ud835\udc5d)..........proved"
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