“Finally we settled on 'Recordbreaker'.”
You're a member of a band. You're telling someone the story of how you chose the name for you band. You told the set-up for your story, and now you're telling the result. You say this.
Finally we settled on 'Recordbreaker'.
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"Finally" is an adverb that tells when you did something. It means that you did it at the end of a long process.
"Finally can come in several places in a sentence:
Finally we settled on "Recordbreaker".
We finally settled on "Recordbreaker".
The most common location for "finally" in casual speech is at the beginning of the sentence, as in the first example above. In written English, it usually comes just before the verb, like in the second example.
To "settle on" something means to choose it after a debate or negotiation. In the example at top, if the band "settled on" the name "Recordbreaker", that probably means that some of the band members wanted to use a different name.
To "break a record" means to get a higher score or a greater accomplishment than anyone else in history. For example:
Phelps broke seven world records at the 2009 Olympics.
This means that Michael Phelps swam faster than anyone in history for 7 different events.
Bands often like to come up with unique names that don't really mean anything. So the meaning of "Recordbreaker" (a person who breaks records) doesn't have to relate to the band or its music.