Actually, why they looked up to her is because Ariel actually pursued her dreams, and was willing to go for any cost to achieve them. And for the record, no, Ariel did not only desire to become a human just because of Prince Eric. Actually, compared to the other Disney Princesses before her aside from possibly Cinderella, Prince Eric played the least (though certainly not insignificant) role in Ariel's desire to become human, as she already harbored a fascination with humanity and desire to be among them long before she even met Eric. In fact, all Eric did was give her the final push. And she actually did get to know him so to speak by observing his actions on the ship. And keep in mind, that was BEFORE Eric was even aware of her existence. Other reasons why she is a good role model is that she actually does value her friends lives above even her own, even if she has every reason to let them die (she actually rescued Flounder when he was nearly eaten by Glut, putting herself at great risk in the process, and that's not even getting into her saving Sebastian from becoming Grimsby's meal), she actually owns up to her mistakes, and was actually demonstrated to be compassionate, and she's not afraid to pursue her dreams either, and she actually harbors a moral compass and a conscience. Oh, and for the record, she didn't "give up her life." And BTW, that's just in the original film.
In fact, if any of the Disney Princesses acted as a truly bad role model, it's either Belle or Merida.
For Belle:
In her debut film she basically managed to be, despite intending to be somewhat pure of heart, not even coming close to it and in effect ruining it. For starters, she basically talked very badly about her own home and arrogantly felt she was far more superior to the village for little reason; she basically behaved like a hypocrite by decrying Gaston as being rude and conceited; she made the jerk move of making it seem as though she actually would say yes to Gaston and then basically throwing him into the mud and then waving goodbye mockingly and with some amusement, and to make matters worse, she wasn't even being forced to marry at that time (at least Jasmine had the fact that she was forced to marry by law as an excuse for her actions, Belle doesn't); she behaved like a total jerk to the Beast even after he at least tried to make amends for his earlier behavior to her father by, you know, actually giving her a room and relatively free access to the castle; she deliberately disobeyed Beast and the servants' strict warning about not going into the West Wing (and based on her tone when finding out what the West Wing was, it was out of spite, not curiosity); she nearly and quite stupidly nearly destroyed the Beast's lifeline, that enchanted rose, even though the fact that it was underneath a glass belljar should have made it very obvious that it was not something meant to be touched [something that Belle clearly should have figured out since, you know, she DID deduce from casual observation that the castle was enchanted not a few moments before]; she then broke her end of the agreement by trying to flee the castle, not to mention do so during a snowstorm where she clearly hadn't taken into account the wolves and the elements, nearly getting Beast killed trying to save her butt, and she still blames HIM for the situation even when that situation had largely been Belle's fault (at least Ariel actually owned up to the mistakes she made and didn't blame her father or anyone else for them, not even in instances where her father actually was at fault for it); there's little indication that she even liked the Beast, even after he saved her, until after he gave her a library, which frankly falling for someone for their library is far worse than falling for them for their looks; fortunately she got a bit better by the second half, but then she basically stupidly exposed Beast to a congregated, rowdy, drunk-on-emotions mob that was led by her suitor in a clear attempt at blackmailing her, and this was despite figuring out beforehand that Gaston was trying to blackmail her, and then she gets all surprised when Gaston riles up the mob and has them try to kill the Beast, with the servants being collateral damage, for reasons clearly out of jealousy, even when it should have been extremely obvious even to a preschooler what would happen as a result? In fact, with her actions, you could argue she promoted that you can sell out your friends and do very bad things and even if you don't really own up to it you still get what you want. Oh, and I wouldn't even say she had much book smarts, especially when the only thing in the film she actually seemed to read are fairy tales, not really stuff an advanced reader tends to read (they would have had her reading Romeo and Juliet to Beast had "Human Again" not been cut out). And in Disney Comics, it's even implied she's a misandrist. Thankfully, Disney Comics aside, the future installments for the franchise (including the Marvel Comics, the DTV midquels, books, and all of that) do actually try to make sure she actually matches up with purity of heart, but still...
And as for Merida:
She basically behaved like a man for most of the film for no real reason other than the film itself promoting radical feminism/misandry; She basically ruined a marriage ceremony despite the fact that, while technically arranged marriages, she actually was given a choice of who to marry, which is definitely more than what Jasmine had to deal with for most of her film; and she basically tried to drug her own mom and went to a witch specifically to screw with her.
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