And now we come to one of the nastiest things in this story. Up to that moment Edmund had been feeling sick, and sulky, and annoyed with Lucy for being right, but he hadn't made up his mind what to do. When Peter suddenly asked him the question he decided all at once to do the meanest and most spiteful thing he could think of. He decided to let Lucy down. (5.4)
Edmund's first betrayal is a small but unpleasant one – he lies about his trip to Narnia in order to make himself look superior in the eyes of Peter and Susan. This small cruelty will pave the way for his greater treachery later on.
Mr. and Mrs. Beaver
"I didn't like to mention it before (he being your brother and all) but the moment I set eyes on that brother of yours I said to myself 'Treacherous.' He had the look of one who has been with the Witch and eaten her food. You can always tell them if you've lived long in Narnia, something about their eyes." (8.61)
Like most people in Narnia, Edmund wears his heart on his sleeve. OK, well, nobody falls in love here, but you know what we mean – he wears his thoughts on his face. Appearances are rarely deceptive in this world, and if Peter, Susan, and Lucy had known what to look for, they might also have realized what was happening to Edmund.
"The reason there's no use looking," said Mr. Beaver, "is that we know already where he's gone!" Everyone stared in amazement. "Don't you understand?" said Mr. Beaver. "He's gone to her, to the White Witch. He has betrayed us all." (8.53)
Mr. Beaver isn't prejudiced by a family relationship to Edmund, and he's able to recognize right away that Edmund's sudden disappearance means trouble and danger.
And Edmund stood in the shadow of the arch, afraid to go on and afraid to go back, with his knees knocking together. He stood there so long that his teeth would have been chattering with cold even if they had not been chattering with fear. How long this really lasted I don't know, but it seemed to Edmund to last for hours. (9.9)
It takes a great and awful effort for Edmund to take the last steps through the Witch's courtyard and deliver his siblings into her hands. Unfortunately, he goes through with it.
You mustn't think that even now Edmund was quite so bad that he actually wanted his brother and sisters to be turned into stone. He did want Turkish Delight and to be a Prince (and later a King) and to pay Peter out for calling him a beast. As for what the Witch would do with the others, he didn't want her to be particularly nice to them – certainly not to put them on the same level as himself – but he managed to believe, or to pretend he believed, that she wouldn't do anything very bad to them (9.3)
The narrator is careful to explain that Edmund isn't a completely evil being. He's just allowed some of his more negative qualities, like selfishness, greed, and constantly feeling ill-treated by his brother and sisters, to affect his actions disproportionately. He lies to himself about the real consequences of his actions.
Edmund meanwhile had been having a most disappointing time. When the Dwarf had gone to get the sledge ready he expected that the Witch would start being nice to him, as she had been at their last meeting. But she said nothing at all. (11.1)
We really shouldn't be surprised that the Witch betrays her promises to Edmund almost as quickly as he betrays his family.
"But where is the fourth?" asked Aslan.
"He has tried to betray them and joined the White Witch, O Aslan," said Mr. Beaver. (12.17-18)
It's interesting that this conversation takes place. Aslan pretty much knows everything that is going on in Narnia, and he definitely knows what's up with Edmund and the Witch, so we assume that asking about the fourth child is just a formality. It needs to be said outright that Edmund is a traitor.
The White Witch
"You have a traitor there, Aslan," said the Witch. Of course everyone present knew that she meant Edmund. But Edmund had got past thinking about himself after all he'd been through and after the talk he'd had that morning. He just went on looking at Aslan. It didn't seem to matter what the Witch said. (13.37)
Edmund's conversation with Aslan dispels all the after-effects of his betrayal. Edmund has begun to change radically and forever, and part of that change is that he's not thinking about himself all the time.
The Witch was just turning away with a look of fierce joy on her face when she stopped and said,
"But how do I know this promise will be kept?"
"Wow!" roared Aslan half rising from his throne; and his great mouth opened wider and wider and the roar grew louder and louder, and the Witch, after staring for a moment with her lips wide apart, picked up her skirts and fairly ran for her life. (13.54-56)
Because the White Witch is herself a treacherous creature, she assumes that Aslan will try to trick her if he can. Of course, we know he would never do anything like that – and he doesn't. It's not in his nature.
"You know that every traitor belongs to me as my lawful prey and that for every treachery I have a right to a kill." (13.41)
The existence of betrayal in Narnia is what gives the Witch a basis for her power.