Ignorant writer. There are an infinity of infinities and they are of distinct cardinalities. The infinity this writer is probably talking about is the number of whole numbers, which is the smallest infinity. For this infinity, that statement is false because the number of elements in a set with "infinity" elements can be placed into one-to-one correspondence with a set containing the same elements, but with one additional element added.
There are different types of infinities as well. The number of points on a line segment greatly exceeds the number of whole numbers, and this can be shown fairly easily. For more information, see any good undergraduate text in real variables.