This is my personal favorite couple of the series, they are so cute together and they just fit. I love how Renton is at a total loss for words (for the most part) around her and how through Renton, Eureka is learning how to be more human.
It's obvious that they love each other deeply, heres some evidence.
In the very first episode, after the military attacks and Eureka and Holland go to fight them, Renton's grandfather gives him the Amita Drive, Renton then risks his life to deliver it to Eureka, in the confusion of him arriving on the battle field, Eureka is knocked out, Renton, taking the controls of the Nirvash somehow executes the Seventh Swell Technique, only after he is brought onto the Gekko and questioned by Holland does he reveal that.
Renton: In that moment... All I wants in the world... Was to protect Eureka, nothing else mattered.
He risked his life to protect her, I call that love at first sight.
Then, a few episodes later, after a fight with Holland, Renton leaves the Gekko to find himself. Later Eureka, wondering where he's gone, finds out that he's left, feeling that SHE was the reason she sets off without telling anyone to find him, coincidentally the military attacks the Gekko, Renton returns to the ship just as the attack commences, after a quick scolding by Holland, he asks to see Eureka, not finding her he takes the Nirvash out to find her. After saving her from a fall and initiating a more powerful version of the Seventh Swell, they slowly make their way back to the Gekko. This is the conversation they have.
Renton: What... ever made you want to do something so dangerous...?
Eureka: I... I wanted to see you again.. I...
Renton: I know... I wanted to see you too...
Eureka: I've wanted to talk to you... For so long...
Renton: I.. .want to know everything thats happen... and how everyone has changed.... I'm... Never going to leave you ever again
Eureka: *smiles and looks up to him blushing lightly*
*They hug*
Later on the two start opening up their feelings for each other, even the kids start accepting Renton as a constant presence at Eureka's side, heres are some possible spoiler quotes:
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Renton: I'm sorry, Eureka. I have to defeat some of your fellow Coralians, and if you tell me that's a sin, then I'll just have to carry its weight on my shoulders, because no matter what, I HAVE TO GO WHERE YOU ARE, EUREKA!<==
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Maurice: Nirvash finally answered your calls, didn't he?
Renton: That's right, Maurice.
Maurice: Please, you got to go and save Mama! Our Papa is the only one in the world who can save our Mama!<==
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Eureka: I have people that are dear to me. Linck, Maeter, Maurice and Renton. I like them a lot. I love them. People dear to me, my family. They taught me all sorts of things when I didn't really know anything. They accepted everything about me before even I did. They let me be who I am. I want to be with those that I love, that I want to protect. I want to be with Linck, I want to be with Maeter, I want to be with Maurice, and, of course, Renton. But if wishing so causes those that are dear to me to be lost, if wishing so makes the planet where they all live disappear, I will stop wishing. But if it can be allowed, I would like to see them one more time. I want to see you. I want to see you... Renton!<==
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So yeah.... The quotes may not make much sense... But yeah, they are made for each other, I support them 110% of the way.
-Tibs