Basic Real Analysis by Anthony W. Knapp

By Anthony W. Knapp

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If the series is CEl ak ( x ),one wants C ; = , a k ( x ) to be small for all m and n sufficiently large. Some of the standard tests for convergence of series of numbers yield tests for uniform convergence of series of functions just by introducing a parameter and ensuring that the estimates do not depend on the parameter. We give two clear-cut examples. 18. 19. 18. If for each x in a nonempty set S, { a , ( x ) } , > l is a monotone decreasing sequence of nonnegative real numbers such that limn a , ( x ) = 0 uniformly in x , then CEl (- l ) " a , ( x ) converges uniformly.

11; The next topic is the problem of interchange of integral and limit. EXAMPLE. On the interval [0, 11, define f,(x) to be n for 0 < x < l / n and to be 0 otherwise. 27 allows us to see that Jb f, dx = 1for all n . On the other hand, limn f, (x) = 0 1 1 forallx E [0, 11. Since& Odx =O,wehaveJb f,dx = 1 # O = l i m , l f,dx. Thus the interchange is not justified without some additional hypothesis. 31. If { f,} is a sequence of Riemann integrable functions on [ a ,b ] and if { f,} converges uniformly to f on [ a ,b], then f is Riemann integrable on b [ a ,b ] ,andlim, lof,dx = lob f dx.

Fur the asserted f dx 5 inequality we have f 5 If 1 and - f 5 I f 1, so that (c) and (b) give lab If 1 dx and - Jab f dx 5 J~~ 1 f 1 dx. Combining these inequalities, we obtain lab I ~ ~ ~ ~5J)2Dlfldx. f d x l For (f), the first conclusion follows from (d) with rp(t) = t2. For the Riemann integrability of f l f2, we use the formula f l f 2 = +(( fl f212 - f; - f22) and the earlier parts of the proposition. Conclusion (g) follows from (d) with rp(t) = ,,h. For (h), each partition P of [ a ,b ] yields a natural partition P' of [-b, -a] by using the negatives of the partition points.

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