Draw the real line and label from about -5 to 5 in increments of 1. There are an infinite amount of integers on the real line. Based on your sketch you'll also agree with me that there are twice as many total integers as there are positive integers, right? But how many positive integers are there? That's it... an infinite amount. So there are infinite amounts of integers both on the entire real line and on each half of the real line. Are you scratching your head yet?

Now consider the real numbers. In fact, lets narrow our focus way down to just the part of the real line between 0 and 1. Circle that part of the real line. How many real numbers are there in between 0 and 1? What can you conclude from all this?