I Upper and Lower Darboux Sum Inequality

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TL;DR Summary
L(f,P) ##\leq## L(f,Q) ##\leq## U(f,Q) ##\leq## U(f,P)
Lemma
Let f be a bounded function on [a,b]. If P & Q are partitions of [a,b] and P ##\subseteq## Q , then

L(f,P) ##\leq## L(f,Q) ##\leq## U(f,Q) ##\leq## U(f,P) .

Question is "How can P have bigger upper darboux sum than Q while it is a subset of Q"
 
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The finer partition will be as close or closer in both upper and lower sum to the limit than the courser partition. For the upper sum, the supremums of the courser partition are still there, but they are applied to smaller ##\Delta x##s.
 
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The finer partition will be as close or closer in both upper and lower sum to the limit than the courser partition. For the upper sum, the supremums of the courser partition are still there, but they are applied to smaller ##\Delta x##s.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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