At the end of the narrative, Jerry agrees to accompany his mother because he has already proven to himself that he can do what the older boys do; he has swum through the tunnel and completed his rite of passage.
Once he has gone through the tunnel of rocks and is able to hold his breath successfully for two to three minutes, Jerry has proven to himself that he is the equal to the older boys. For, he has conquered the mental and physical challenges of swimming under water for minutes. By having overcome these challenges, Jerry senses his maturity, and no longer feels that he has to prove anything. So, with his new sense of independence and greater maturity, Jerry welcomes his mother's suggestion that he swim no more in one day.
...he gave in at once. It was no longer of the least importance to go to the bay.
Having earned his rite of passage, Jerry is ready to rest.