On Uniform Convergence Page: 8
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Sf^x)-* f(x), en ^a, b], i.e., lim F (x) ^ f(xh is that
. 33-nM,
for every e > 0 and for x in [a, b] thero exists an H auch
that, for n > N and m > N,
!Rg(x) * P^(x)f <' 8.
1*19* ^ cmd If for every n,
!^nf ** %i*
then eoBveygga*
1.20. Ea^ eenverKaa if aaa4 only if for every e > 0
they# exists an N suoh that for n >
j f ^ ^ *
whoro is tho nth remainda-r. of
§$+ Senary of Chapters
1.21. As we have aeen, the purpose of Chapter I is
two-fold^ The first being to aaqpaiat the readep with the
eonaopt of uniform 3env@rg^aee% and the s#oond.^ a statement
of assumptions* Now, in studying the property of uniform
convergence, it might not b$ convenient or ytraetieal to
alwayg use th@ definition giv^n in Chapter I. Further,
it might not b<$ posaibl# to toll if a sari#3 is uniforaly
eenvergont without ether definitions and t^sts. This 1#
the reason for Chapter II. In it the reader will find #th@r
definitions of uniform convergence and statements of aoma
of the mere important tests listed in it aystem&tlo -Banner.
Chapter III is an investigation of the behavior of uniform
convergence* In it several fundamental theorem# (concerning
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Drew, Dan Dale. On Uniform Convergence, thesis, February 1951; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc96930/m1/11/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .