If there is a regular language $M$ over $\Sigma = \{0,1\}$, then show that language $$L = \{x : x\in M \text{ and $x$ ends in 10}\}$$ is regular as well. I'm thinking of closure under Union or Difference but cannot connect the dots.
Really appreciate any suggestions.
HINT: The intersection of two regular languages is regular. In case you don’t already have this result available, you can prove it by starting with finite state machines accepting the two languages and combining them to get a finite state machine that accepts their intersection. The combination uses what is often called the product construction, since the state set of the new machine is the Cartesian product of the state sets of the original two machines.