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Would this thinking be correct?

The traveller is travelling infinitely close to the speed of light, since he makes a round trip (changing direction and returning home), the total time for the traveller will be less than the time elapsed on earth, therefore causing the traveller to have aged less than the observer.

The closer to the speed of light the traveller travels at, the less time elapses for the traveller in one round trip, until the time elapsed is infinitely small, and to an observer who has aged any amount of time (depending on how many round trips the traveller makes) it appears that no time has elapsed for the traveller.

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