n
 E |A fI l E IA[, f| > a ,
 R P Y i=1 [(s ,t i-1) ( ,t0l' ,i)] 2

 The rectangles in this sum form a partition of the n rectangles

 [(rl,t),(p,t0, )], [(r2',t 1),(P 0,2)], ... [(rn,tOn-1),(p,t')]

 (shaded rectangles in Figure 2-7). Thus


 n
 E Var (f)
 i-1 [( 't0,i-1 0, l

 n
 > E A f E [(s, t ) )f I > -
 R EP Y i=1 0,1-1 ) (rl 0,


 In particular, then, Var > a Let p = By
 [(r ,t),(pt')] 2 1 n


assumption, Var [',t),(r t f)n > a, so we can repeat this

procedure and get another point s' < r' < r such that
 n n
Var[(r, t),( ,t.)](f) > a Continuing in this manner, we can
 4
 n 1
construct a sequence p0, p1' P2, such that

 i) P > Pi+ > s, pi*s

 ii) Var. t )(f) > a for all i.
 L(p i+ t),(Pit )] 1+1

We have, then, by additivity,


 i-1
 Var (f) = Z Var (f)
 L(pi t),'(Pot )]f) J=O 1(pj+ t),(pj1t')

 i-1 i-1
 > E (a -- ) = ia E --- > a- .
 j=0 23 j=0 2 +1

 1-1
Then Var (f) > E Var (f) > ia-E.
 2 j=0 Pj+1't)' t')J