Define a countable set. use your definition in conjunction with appropriate results to prove that the union of finitely many countable sets is necessarily count

Define a countable set. Use your definition in conjunction with appropriate results to prove that the union of finitely many countable sets is necessarily countable. Use L'Hospital's Rule to compute the limit lim {(1- sinx)/(x sinx)}. Explain briefly the steps followed. x~0+ For any subset S of a metric space X, define S°= the Interior of S. Show that S°is an open subset of S. If S and T are any two subsets of the same metric space X, show that (SnT°= S° nT°.