Infinity is a difficult concept. You are asking: why is 0.999 . . . with
an infinite number of 9's, equal to the integer 1? Well, it is just a
different way of representing the same number. It even makes sense logical-
ly. Remember how you learned how to count using apples? Let us go back to
that concept. Say you have 9 apples and you give one to a friend. Now what
fraction of the total number of apples have you given away? Right, 1/9.
How do we represent that in decimals? Right, 0.11111 (repeating). Try it
by hand. You will quickly see that the 1's go on forever. Now, how many
apples does your friend have? She has:
(total number of apples) times (her fraction of the total number) =
= (9) times (0.11111 (repeating)) = .99999 (repeating) = 1 apple.