How does Ariel prove to be a loyal servant?

Loyal Arie. Ariel is Prospero's spirit (he's not human) servant who manipulates others by changing shapes as Prospero tells him. Ariel is loyal to Prospero because Prospero rescued Ariel from imprisonment in a tree by a witch.

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One may also ask what does ariel symbolize in the tempest?

Early in the play, Ariel reports the plot to murder Prospero, and later, he assists in punishing Prospero's enemies. Ariel's obedience is an important symbol of Prospero's humanity, because he ameliorates Prospero's role on the island and humanizes the action that Prospero takes against his old adversaries. Why does Ariel first speak of mercy? Ariel was the first to speak of mercy because he wanted to be free and he didn't want to do Prospero's deeds any more (although he was willing and enjoyed doing it). Prospero decides to give up magic, because magic (and studying magic) is what caused him to end up on the island in the first place.

And another question, how will you prove that ariel is a delicate spirit loved by prospero?

Unlike Caliban, Ariel has a (mostly) warm and loving relationship with Prospero, who saved Ariel when he arrived on the island. (The evil witch Sycorax imprisoned Ariel in a tree because the "delicate" spirit didn't have the heart to do her bidding.) One may also ask who was sycorax what did she do to ariel and why? She was born and banished from Argier (the older form of Algiers). She is mentioned in 1.2 and 5.1. Ariel was her servant but punished him for disobedience (Ariel was "too delicate/To act her earthy and abhorr'd commands"), trapping him in a cloven pine for a dozen years, during which time she died.

What did Ariel Sycorax do?

Prospero declines, reminding him of the state he was in before Prospero rescued him: Ariel had been trapped by the witch Sycorax in a "cloven pine" as a punishment for resisting her commands.