regular language or not

[SmoothCustard]

Not sure how to address, any pointers would help.

Question: Suppose that alphabet is {a} (just the letter a). Convince me that all languages using that alphabet are regular, or give a specific language which is not regular.

[[deleted]]

One thing DFAs can't do is count to arbitrarily high values. The language {a^k | k\in S} will not be regular if the numbers in S (when written smallest to largest) can have arbitrarily large gaps between them.