**Explanation:**

The set {2x | x is any positive rational number} is an infinite set. Here's why:

**Definition of an Infinite Set:**
An infinite set is a set that has an uncountable number of elements. It cannot be exhausted or enumerated completely.

**Definition of Positive Rational Numbers:**
Positive rational numbers are numbers that can be expressed as a fraction of two positive integers. For example, 1/2, 3/4, and 5/6 are all positive rational numbers.

**Definition of Set Builder Notation:**
The set {2x | x is any positive rational number} is defined using set builder notation. This notation states that for each positive rational number, x, we can obtain an element in the set by multiplying x by 2.

**Proof that the Set is Infinite:**
To show that the set {2x | x is any positive rational number} is infinite, we need to demonstrate that there is no limit to the number of elements in the set.

Let's assume the opposite, that the set is finite. This means that there is a finite number of positive rational numbers, which we know is not true.

Since positive rational numbers can be expressed as fractions, we can always find a new positive rational number by adding 1 to the denominator of a fraction. For example, if we have 1/2, we can get 1/3, 1/4, 1/5, and so on, by adding 1 to the denominator.

Since there is no limit to the number of positive rational numbers, there is also no limit to the number of elements in the set {2x | x is any positive rational number}.

Therefore, the set {2x | x is any positive rational number} is an infinite set.