The question is in Wade's "Introduction to Analysis" 3rd. edition. Question 1.3.4 Prove that for all real numbers 'a' and natural numbers 'n' there exists a rational number r_n such that |a - r_n| < 1/n. (my) Proof: |a - r_n| < 1/n iff -1/n < a-r_n < 1/n iff -1/n - a < -r_n < 1/n - a iff a - 1/n < r_n < a + 1/n Certainly, a - 1/n < a + 1/n, and both are real numbers by closure properties of R. Since I already proved that the rational numbers are dense in R, there exists a rational number between a - 1/n and a + 1/n, and we are done.// Is this a valid proof or am I missing something? If it is correct, it seems that this is no more than an alternate statement of the density of Q in R, but I sense that I could be missing a subtler proof or point. Any help is appreciated!