With the Iowa caucuses just 15 weeks away, Mr. Biden acknowledged the result of his protracted deliberations, saying that a “window” had closed on “mounting a realistic campaign.”

“Beau was our inspiration,” Mr. Biden said, with his wife, Jill, at his side. “Unfortunately, I believe we are out of time.”

Beau was the second child Mr. Biden had lost: His daughter and his first wife died in a car accident decades ago.

At the wake for his son, Mr. Biden told friends that Beau, in his final days, had said he hoped his father would run for president. Mr. Biden shared that story repeatedly in the weeks to follow. It struck many who heard it as a form of therapy for a grieving man.

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