Take a piece of string and cut it in half. You now have two pieces of string, each half the length of the original. Now, cut one of the halves in half again. Cut one of the remainders into half, and this again, etc. The fact is, no matter how many times you cut the string into half, you will never achieve a state where the length of the segment reaches zero.
So, eventually, you will have cut the string in half an infinite number of times, and hence you will have an infinite number of pieces of string which will all have a length above zero.
Obviously, any value above zero multiplied by infinity equals infinity, and hence through logical deduction we have proved that any piece of string is infinitely long.