Disclaimers: I do not own The Lord of the Rings and The Silmarillion. J.R.R. Tolkien does. The movie verse for "The Lord of the Rings: The Return of the King" belongs to New Line Cinema.
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From the fanfics I've read so far on this subject, concerning what happened inside Mount Doom, it always seems to end with Frodo's death. From The Lord of the Rings wiki, it said that in one of Tolkien's letters that Eru intervened, leading to Gollum's death and the destruction of the One Ring. So there is the source that I found to clarify how I came to this conclusion. But in this instance, there are multiple sources that say this and apparently it can be found in one of Tolkien's letters.
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This was it. The flames from the lava brought about intense heat, and yet outside the mountain brought about an undeniable chill and heat at the same time. But for Frodo Baggins, the thick, curly brown-haired hobbit with bright blue eyes, felt his end near in sight. His pale hand, covered in his own blood, clung for life on a small rock face on the black ledge. Oh, how he wanted it to end. To finish his quest.
The Ring was right there, calling for him to follow. To meet its doom alongside him.
Frodo turned his gaze upwards. There was Samwise Gamgee, the brown-haired hobbit who reached the end of this quest with him. His hand extended down to him to reach. He wanted to reach for the hand, but he couldn't. He was slipping! He was….
"Do not be afraid." A voice called to him in the air. Frodo's eyes widened. It was Eru's voice and his hand was on his, holding him steady even when he was sweating intensely. "Do not let go."
Courage overwhelmed the hobbit. He could do this. He needed to take this final straw. He pulled himself upright, grabbing onto Sam's arm and not letting go. It was Eru who helped pull him up the rest of the way, just as the lava climbed higher and higher towards the two hobbits. Frodo dared not let go of Sam's hand. He wanted to escape, needed to escape Mount Doom and so landed on top of a rock with his best friend.
In all, Frodo had to thank Eru Ilúvatar for helping save his life. He was about to do something dangerous, and yet one thing was clear, even as the words flew out of his mouth with ease, "It's gone. It's done." The quest was over. The weight was over. He breathed with relief and peacefulness. The last he knew and remembered was climbing up the rock and landing on his back against the rock's smooth surface. He was relieved the quest was over. He was relieved to not have to go through that trauma again. At least, for now. For now, the quest was over and done, and he was spent. He was spent.
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Thanks for reading. :)