»How did Morpheus find out that Neo is the One?«

At the beggining of the first Matrix movie where Cypher talks with Trinity i think Cypher says that if Neo turns out not being the One they (Morheus team?) are going to kill him just like the others.. So if that is correct Morheus in his search for the One also did examine other potential candidates (like the kids in the Oracle's house, kid with spoon etc.) but after he found out they were not the One (how exactly?) he killed them? Why kill and not set them free like they have done with all the other people they have freed and why the candidates at the Oracle's house were not killed since Neo was found?

Maybe from forums? In the original movie scripts they meet in forums

dtragoud wrote:

At the beggining of the first Matrix movie where Cypher talks with Trinity i think Cypher says that if Neo turns out not being the One they (Morheus team?) are going to kill him just like the others..

I believe Cypher says that if he isn't the One, he's going to get killed,
"just like the others"

I think the 'others' (the potential Ones) where also told by Morpheus that they (could be/) were the One.
Maybe after that the ones before Neo just got just a little too much overconfident. Trying to do everything "The One" should be capable of.
But, since they didn't have that powers, they could have got killed by an agent.