What proof do the Tucks give that they won't die in Tuck Everlasting?

The Tuck family tells their story to Winnie: 87 years ago, they drank from a spring while passing through a wood and marked that place with a "T" in the tree's bark. Angus, Mae, Miles, Jesse, and even their horse drank the water, but not their cat. After that, a number of odd things happened. Jesse fell on his head but was fine. The horse was shot by hunters yet was unharmed. As people grew old around them, they realized they were not aging. Eventually, the cat died. Tuck provided proof by shooting himself, which showed that the bullet went through him without harming him. The piece of proof that linked their immortality to the spring was the fact that the tree had not changed while the rest of the woods had. The "T" looked freshly carved.
Winnie is reluctant to believe this story, but the Tucks seem like genuine, friendly people. Tuck takes her out in the rowboat to try to demonstrate the wheel of life. The next morning, Miles tries to convince Winnie to keep their secret by making her imagine what the world would be like if mosquitoes never died.
Another piece of proof doesn't come from the Tucks, but the man in the yellow suit. He tells the Tucks that a friend of his grandmother talked of marrying into a family that never aged.

Winnie found that she could scarcely breathe. It was true, then! Or was the man who stood there crazy, too?

After the commotion with the man in the yellow hat, Winnie realizes the terrible truth.

Whatever happened to the man in the yellow suit, Mae Tuck must not be hanged. Because if all they had said was true, then Mae, even if she were the crudest of murderers and deserved to be put to death—Mae Tuck would not be able to die.