"It's just really hard to believe."
"What? No! I came all this way! Please, I swear I can do magic, please!"
"Then show it."
"I... I don't know how!"
"Adara," Ruby says, before she freaks out enough to summon another Atronach. Not that that wouldn't be a good enough demonstration, probably, but it's better if it's under her control. "Breathe. Close your eyes. Look at the frustration and indignation you were just feeling, and find the place within you it was starting to draw from, the same place you have used magic in the past from."
"Think on the times you've cast spells, in the past. Visualise the Daedra you've been summoning, the elemental spirits beyond the veil of Oblivion. Offer them your magicka, and build them a body. There is plenty of ice-aspected magic around here, feel that, too. Even if you cannot draw on it, allow it to ease the path of the Atronach."
"A Frost Atronach?" Faralda observe. "That is impressive, if true."
Faralda's words tick Adara off again, and she can find it, the place within her her feelings want to use. These instructions are a lot more specific and legible than anything she's done before, but she can use her own intuition to help her along and...
...conjures a Frost Atronach.
"Well, then. Colour me surprised," says Faralda. "What is your name, child?"
"I'm Adara, ma'am."
"I believe you will make a fine wizard some day. Can you have your Atronach step onto that sigil on the floor? That way I will be able to key your magical signature into the wards of the College so you can walk in."
She's never actually been able to control her conjured creatures but she tries just—thinking at it—and it works, and she beams with pride.
Faralda casts another spell into the sigil then says, "Welcome to the College, young lady. I'll let Ruby show you in."
Ruby teaches her how to dispel her Atronach and then leads the way up the bridge and into the main gate of the College. "One of the staff will probably give you a proper tour, but that tower there is the Hall of Attainment, which is where you'll probably sleep, and the main tower over there is the Hall of Elements, where most lessons are held and the staff have their offices. I'll take you to Mirabelle—she's the College's, ah, second in command, so to speak?—and she'll take care of your admissions process."
She can take care of that, sure. Adara is nearly vibrating out of her skin, which Mirabelle finds very endearing, and she dismisses Ruby.
Who finds Sergius to explain what happened.
Or to try to explain what happened.
"What? Oh, that sword. I don't care, I'm busy."
"...so you don't want the money?" Not that Ruby has any but he assumed the sword was being sold.
"No, you can keep it. The joy of enchantment is enough payment, blah blah blah, now leave."
Sergius is so very himself. But, well, all's well that ends well, Ruby guesses.
So now he wants to study and practise. He can't take the Valtheim Towers as he is, he'll need to learn how to swing a sword and shoot a moving target if he wants to stand a chance.