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suppose that the following relation holds. 

Then, invoking (14) the relation in (33) can also be expressed as 

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The relation in (35) indeed holds as a result of (34) and the fact that Γ ≤ ˜ Γ = Γ + ˜ δ and that the slope curve S ( z, ˜ Γ ) is non-increasing in z . Notice that the integral in the LHS of (35) is the area covered under rectangles of non-increasing heights S ( z, ˘ Γ ) , z ∈ [ ˜ Γ , ˜ Γ + δ ] and total width δ . An analogous observation holds for the RHS of (35). It remains to prove for this case that (34) indeed is true. We accomplish this via contradiction. and suppose that ∃ Z ′ ≥ ∑ b ∈B ′ 1 ¯ h ( ˜ Γ$_{b}$ , b ) : S ( Z $^{′}$, ˘ Γ ) > S ( Z $^{′}$, ˜ Γ ) . Then, notice that ∑ b ∈B ′ 1 ¯ h ( ˜ Γ$_{b}$ , b ) ≥ ∑ b ∈B ′ 1 ¯ h ( ˘ Γ$_{b}$ , b ) . Consider next how macro resources are assigned by Algorithm I for given pico budgets ˘ Γ , ˜ Γ . We can see that for both pico budget vectors, the macro resources to all TPs in B ′ 1 \ b$_{2}$ are assigned as slack in the same order, i.e., if a TP b ̸ = b$_{2}$ is assigned a macro resource share for the n th time (for any n ≥ 1 ) under ˘ Γ then that TP can be assigned a macro resource share for the n th time under ˜ Γ only after all prior macro assignments made to TPs in B ′ 1 \ b$_{2}$ under ˘ Γ have been made to those TPs under ˜ Γ as well. Then, if upto Z ′ 

• At-least as much macro resource has been assigned as slack to TPs in B ′ 1 \ b$_{2}$ under ˘ Γ than under ˜ Γ , we immediately have the contradiction since S ( Z $^{′}$, ˜ Γ ) must be at least as large as the slope value at the most recent assignment to made to any TP in B ′ 1 \ b$_{2}$ under ˘ Γ . The latter slope value in turn must be no less than S ( Z $^{′}$, ˘ Γ ) , since S ( Z, ˘ Γ ) is non-increasing in Z . 

• Less macro resource has been assigned as slack to TPs in B ′ 1 \ b$_{2}$ under ˘ Γ than under ˜ Γ , so that at-least ¯ h ( ˜ Γ$_{b}$$_{2}$ , b$_{2}$ ) − ¯ h ( ˘ Γ$_{b}$$_{2}$ , b$_{2}$ ) more macro resource as slack has been assigned as slack to TP b$_{2}$ under ˘ Γ . Invoking (29) (for b = b$_{2}$ ) we again immediately have the contradiction since S ( Z $^{′}$, ˜ Γ ) must be at least as large as the slope value at the most recent assignment to made to TP in b$_{2}$ under ˘ Γ . The latter slope value in turn must be no less than S ( Z $^{′}$, ˘ Γ ) since S ( Z, ˘ Γ ) is non-increasing in Z . 

