alt="parallel-to f Subscript n Baseline left-parenthesis t 0 plus delta right-parenthesis minus f 0 left-parenthesis t 0 plus delta right-parenthesis parallel-to less-than epsilon and parallel-to f Subscript n Baseline left-parenthesis t 0 Superscript minus Baseline right-parenthesis minus f 0 left-parenthesis t 0 Superscript minus Baseline right-parenthesis parallel-to less-than epsilon period"/>
When
On the other hand, when
But this yields the fact that
The next lemma guarantees that, if a sequence of functions
Lemma 1.15: Let be a sequence of functions in . Suppose, for each , the function satisfies
for every , where for each and the sequence is equiregulated. Then, the sequence is equiregulated.
Proof. Let
A clear outcome of Lemmas 1.13 and 1.15 follows below:
Corollary 1.16: Let be a sequence of functions from to and suppose the function satisfies condition (1.2) for every and , where and the sequence is equiregulated. If the sequence converges pointwisely to a function , then it also converges uniformly to .
1.1.4 Relatively Compact Sets
In this subsection, we investigate an extension of the Arzelà–Ascoli
