Proving a Language to be regular

Question

Pumping Lemma is used to prove a language to be not regular. But How a language can be
proved to be regular ? In particular,

Let L be a language. Define half(L) to be  
{ x | for some y such that |x| = |y|, xy is in L}.  
Prove for each regular L that half(L) is regular.  

Is there any trick or general procedure to tackle such kind of questions ?

Answer 1

If you can correctly describe your language L by an NFA or DFA, then it will be regular.

There is a well known equality of NFAs, DFAs, regular grammars and regular expressions, so a representation of L in any of these formalisms should do.

Answer 2

Provide a regular grammar or a finite automaton that matches the language. For the full list of properties you can prove to show a language is regular, see the first lines of the Wikipedia Article on regular languages.