Who is asking: Teacher
Level of the question: Secondary
Question: If a coin is flipped n times, where H is the number of heads after n flips, and T the number of tails, then will the quantity (H-T) change signs infinitely often as n goes to infinity?
The answer to your question is YES, H - T changes sign infinitely often. But the number of sign changes grows very slowly. The behavior of flipped coins is counterintuitive.
The excellent old probability text by William Feller (my copy is from 1957) devotes Chapter 3 to this question. It is worth reading, because it is both elementary and quite surprising. If you cannot find a copy of the text, then perhaps other texts now discuss it -- look under "fluctuations in coin tossing and random walks", or maybe under the arc sine law. Perhaps you can find an appropriate web page. Feller states
...we reach conclusions that play havoc with our intuition. It is generally expected that in a prolonged series of coin tossings Peter should lead about half the time and Paul the other half. This is entirely wrong, however. In 20,000 tossings it is about 88 times more probable that Peter leads in all 20,000 trials than that each player leads in 10,000 trials. In general, the lead changes at such infrequent intervals that intuition is defied. No matter how long the series of tossings, the most probable number of changes of lead is zero; exactly one change of lead is more probable than two, two changes are more probable than three, etc. ... In very few cases will the lead change sides and fluctuate in the manner that is generally expected of a well-behaved coin.