Since the Mirror of Galadriel could show what might happen in the future, could it have shown Frodo that he would succeed at destroying the One Ring?

Question

I am wondering that if the Mirror of Galadriel could show Frodo what might take place in the future, could it have shown him that he would succeed in his quest to destroy the One Ring?

Answer 1

Galadriel says the Mirror does have the power to show the possible future.

‘Many things I can command the Mirror to reveal,’ she answered, ‘and to some I can show what they desire to see. But the Mirror will also show things unbidden, and those are often stranger and more profitable than things which we wish to behold. What you will see, if you leave the Mirror free to work, I cannot tell. For it shows things that were, and things that are, the things that yet may be. But which it is that he sees, even the wisest cannot always tell. Do you wish to look?’ (The Mirror of Galadriel, LotR)

Sam did indeed see at least two things that were to come - the tree-felling was possibly already in progress; the other visions occur when Frodo is stung by Shelob:

But before Sam could make up his mind what it was that he saw, the light faded; and now he thought he saw Frodo with a pale face lying fast asleep under a great dark cliff. Then he seemed to see himself going along a dim passage, and climbing an endless winding stair. It came to him suddenly that he was looking urgently for something, but what it was he did not know. Like a dream the vision shifted and went back, and he saw the trees again. But this time they were not so close, and he could see what was going on: they were not waving in the wind, they were falling, crashing to the ground. (The Mirror of Galadriel, LotR))

However, Galadriel warns Frodo and Sam that changing their actions based on the Mirror is dangerous:

Remember that the Mirror shows many things, and not all have yet come to pass. Some never come to be, unless those that behold the visions turn aside from their path to prevent them. The Mirror is dangerous as a guide to deeds. (The Mirror of Galadriel, LotR)

Frodo seems to have seen only the past, visions of Gandalf and Bilbo, and Sauron searching for him:

Doubt came into Frodo's mind: was this a vision of Gandalf on one of his many lonely journeys long ago, or was it Saruman?

The vision now changed. Brief and small but very vivid he caught a glimpse of Bilbo walking restlessly about his room. The table was littered with disordered papers; rain was beating on the windows.

Then there was a pause, and after it many swift scenes followed that Frodo in some way knew to be parts of a great history in which he had become involved. The mist cleared and he saw a sight which he had never seen before but knew at once: the Sea. Darkness fell. The sea rose and raged in a great storm. Then he saw against the Sun, sinking blood-red into a wrack of clouds, the black outline of a tall ship with torn sails riding up out of the West. Then a wide river flowing through a populous city. Then a white fortress with seven towers. And then again a ship with black sails, but now it was morning again, and the water rippled with light, and a banner bearing the emblem of a white tree shone in the sun. A smoke as of fire and battle arose, and again the sun went down in a burning red that faded into a grey mist; and into the mist a small ship passed away, twinkling with lights. It vanished, and Frodo sighed and prepared to draw away.

But suddenly the Mirror went altogether dark, as dark as if a hole had opened in the world of sight, and Frodo looked into emptiness. In the black abyss there appeared a single Eye that slowly grew, until it filled nearly all the Mirror. So terrible was it that Frodo stood rooted, unable to cry out or to withdraw his gaze. The Eye was rimmed with fire, but was itself glazed, yellow as a cat's, watchful and intent, and the black slit of its pupil opened on a pit, a window into nothing. (The Mirror of Galadriel, LotR)

So, while Frodo did not see his eventual success, and it might have been prevented had it been shown to him, it seems it would indeed by hypothetically possible for the Mirror to show Frodo such visions. We know the Mirror did show true visions of the future, and we have no reason to believe the destruction of the Ring would have been somehow excluded from its capabilities.