Wait, but between n and n+1 there are also an infinite number of rational numbers, but the cardinality of all rational numbers is the same as that of integers (

Wait, but between n and n+1 there are also an infinite number of rational numbers, but the cardinality of all rational numbers is the same as that of integers (or that of all rational numbers between a fixed n and n+1). So, while what the gp wrote is correct, it's not an explanation of why there are more real numbers than integers.