Rationals dense in the reals

I have in my book the statement: Every interval contains both rational and irrational numbers. Now, I have heard somewhere that the rationals are dense in the real numbers, which I assumed was the property stated above, but then it turns out that it means that all real numbers are limit points of the rationals. Are the two statements equivalent.
My guess is yes: For since every interval has rational numbers we can construct at sequence of intersections about any given real number a which by completeness should converge to a. But then choosing the sequence of rational numbers that are in the successive intersections we get a sequence of rationals with limit point a. But my problem is: Do we know that we can always construct a sequence of intersections around any real number?