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By Anthony W. Knapp

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Extra resources for Basic Real Analysis

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FN-1 that the I f,l are uniformly bounded, say by M. Then also If (x)l i M for all x . Put en = sup, I f,(x) - f ( x ) I ,so that fn - E , 5 f 5 f , E,. Proposition 1 . 3 0 and ~ the remark with the proposition, combined with Lemma 1 . 2 5 , then yield + + 1 Hence f d x - / f d x 5 k , ( b - a ) for all n . The uniform convergence of -a { f,} to f forccs E , to tcnd to 0, and thus f is in R [ a ,b]. Thc displaycd cquation, in light of the Riemann integrability of f , shows that b The right side tends to 0, and therefore limn L~f, d x = hbf d x .

PROOF. If n ? m , summation by parts gives Let t > 0 be given, and choose N such that ak ( x ) 5 I f n ? m ? N,then t for all x whenever k > N. and the uniform convergence is immediate from the uniform Cauchy criterion. I. Theory o f Calculus in One Real Variable 20 A third consequence can be considered as a uniform version of the result that absolute convergence implies convergence. In practice it tends to be fairly easy to apply, but it applies only in the simplest situations. 20 (Weierstrass M test).

Let t > 0 be given, let 6 be some corresponding number exhibiting equicontinuity, and choose finitely many rationals r l , . . , rl in [ a , b] such that any member of [ a ,b] is within 6 of at least one of these rationals. Then choose N such that If, ( r j )- f,(rj) 1 < t for + + 24 I. Theory o f Calculus in One Real Variable 1 5 j 5 I whenever n and in are > N. If x is in [ a , b], let r ( x ) be an r j with Ix - r ( x )I < 6. 17. REMARK. The construction of the subsequence for which countably many convergence conditions were all satisfied is an important one and is often referred to as a diagonal process or as the Cantor diagonal process.

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