How to prove using pumping lemma that language generated by a(b*)c(d*)e is regular?
You can't. The pumping lemma can only be used to prove that a language is non-regular. How to prove that it is regular depends on how you've defined regular languages. You (or your course or textbook) might have defined them as any of


languages described by regular expressions;
languages accepted by deterministic finite automata (DFAs);
languages accepted by nondeterministic finite automata (NFAs).


If, additionally, you've proven that some or all of these definitions are equivalent, you can use any of the appropriate ones.

In this case, your language is defined by a regular expression. If that's your definition of regular languages, you're already done.

Our reference question has more information.