Prove that every locally finite measure space is regular.
I am not sure how to go about this problem. Certainly, this can be shown to be true for any ball of finite measure that is contained in a larger ball by local finiteness and continuity of finite measure from above and below, but how would I go about proving this for an arbitrary set $A$?
I don't know where you got this problem from, but it seems to be false, see for example the third example from this section:
https://en.wikipedia.org/wiki/Regular_measure#Inner_regular_measures_that_are_not_outer_regular
You have a locally finite measure there that is inner-regular but not outer-regular(and hence not regular).