Do I have to replace a punctured tire?

Hi,

Last Wednesday (Sept 4), the low tire pressure light came up on the
dashboard. While I had the oil change done, I asked the mechanic to fill the
air. He did and told me that I had two nails, one in each rear tire (wow, what
a good luck). He filled the air and the light was gone. The next day, the
pressure light showed up again. So I went to Discount Tire (where I bought
all 4 tires just 6 months ago) and had them check. They told me that one
could be repaired but the other one had to be replaced. They would have the
tire ready on Monday (Sept 9). They filled the air and the light was gone
again. While waiting for the brand new tire, the pressure light no longer
showed up. It's been almost a week. Does it mean that the tires no longer
loss the air? Is it possible that the nail could just fit in "right" that stops the air
from leaking? It would be great if I don't have to replace the six months old
tire. Please advise. Thank you.