Why did Tuck shoot himself in Tuck Everlasting?

Tuck shot himself to prove to himself and his family, once
and for all, that what they suspect to be the case, is, in fact true. It appears that,
for some reason, the Tucks have become indestructable, and to prove it, he shoots
himself to verify that nothing happens to him.

The Tucks
have realized that something strange has happened to them. There have been a number of
signs - Jesse had fallen from the top of a very tall tree, and "it didn't hurt (him) a
bit." Then, their horse had been accidentally shot, and the bullet had gone right
through him without "hardly even leav(ing) a mark." Then, "Pa got snake bit," and Jesse
ate poison toadstools, and Ma Tuck cut herself, but none of them showed any sign of
damage or hurt. The Tucks became aware that, while their acquaintances were growing
older, they themselves were not, and then, when they return to Treegap, they come upon a
tree that Tuck remembers cutting a "T" into many years before. The tree has not grown a
bit, and the "T" looks as fresh as the day it had been carved. The Tucks remember
drinking from a stream nearby; they and their horse had tasted of the waters, but their
cat had not, and their cat alone had died after a long life some ten years before. The
Tucks conclude that something about the spring water has apparently rendered them
immortal, and to prove it, Pa Tuck shoots himself with his shotgun, and, sure enough,
emerges unharmed (Chatper 7).