I'm trying to identify a science fiction short story I read long ago (I'm 71). The story goes like this:
A human is waiting at a spaceport for his flight to leave. He has been going all around the galaxy looking, for many years, for the secret of immortality.
As he sits waiting, an alien strikes up a conversation with him and they discover they are both searching for the same thing. The alien goes on for quite a while naming hundreds, if not thousands, of places in the galaxy where he has searched.
The Human finally asks the alien how long he has been searching. The alien says "Well, in your years it would be about 15,000 years."
The Alien's flight is called and he leaves to board it.
The human sits for a few minutes, gets up, goes to the ticket counter, cancels his flight and books a flight back to earth. He has realized that if the alien has been searching for 15,000 years, and was very, very old by human standards, what chance did he, a human, have of finding the secret? And, as he figured it, in human years, the alien was awfully close to immortality to begin with.
He had given up the search for immortality and was going home.
That's how I remember the story, so I'm sure I've left things out and changed others. Regardless, can someone please help me with identifying the name for this short story or the author?