Did Rainsford kill Zaroff in "The Most Dangerous Game"?

When General Zaroff went to his bed, Rainsford was hiding
behind the curtain. Then, Rainsford challenged the General. The General accepted his
challenge, adding that the winner would sleep in the bed. At the story's end, Rainsford
claims he had never slept in a better bed. Though not specifically stated, it is implied
that Rainsford killed General Zaroff. Of course, the reader will have to decide based on
what is written:

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A man, who had been hiding in the curtains of
the bed, was standing there.

"Rainsford!" screamed the
general. "How in God's name did you get here?"

"Swam," said
Rainsford. "I found it quicker than walking through the
jungle."

The general sucked in his breath and smiled. "I
congratulate you," he said. "You have won the
game."

Rainsford did not smile. "I am still a beast at
bay," he said, in a low, hoarse voice. "Get ready, General
Zaroff."

The general made one of his deepest bows. "I see,"
he said. "Splendid! One of us is to furnish a repast for the hounds. The other will
sleep in this very excellent bed. On guard, Rainsford." . .
.

He had never slept in a better bed, Rainsford
decided.