Philosophical mathematics question?

I am reading a book about great mathematicians in history and it claims someone solved a paradox I had heard, and I am wondering if I understand it correctly. The name of the paradox escapes me, I don't have the book in front of me, but it says that we should never be able to arrive at a destination, because we first have to travel half the distance, then half the remaining distance, then half the remaining distance, and keep on halving forever, there being an infinite number of halves. I had heard Jodie Foster's character mention this paradox in the movie Contact, but she called it something different than the book I am reading did. It is obviously contrary to real life experience, where people arrive at their destinations, not being trapped in between. The book said this was solved by the existence of irrational numbers on the line that couldn't be expressed as fractions. There are an infinite number of halves, and you could section the line into an infinite number of fractions, but you would have to p points still undescribed by fractions (numbers such as the square root of 2 that have non-repeating decimals) thus the paradox of continuously traveling by halves breaks down. Is this accurate?