“Proof” is what the prosecutors are able to put in front of a jury (based on what they can discover during investigation, the rules of evidence, and the opinion of the trial judge as it relates to admissibility, and despite any objections from the defense) ... and what the jury believes.
If the only evidence is a person saying “I saw him do it,” and if the jury believes that person, then that’s all the proof that’s required. Naturally, a prosecutor won’t want to stake a major case on the say-so of one person, since a good defense attorney can make a single witness look bad in various ways. But when ten people say more or less the same thing, then it’s harder to discredit them all.
Edit: I meant to continue.
Circumstantial evidence has to present a believable story to the jury. Fingerprints on a gun. Gun used in murder. Ammunition in the suspect’s home matches the ammunition used in the crime. Evidence of bad feeling between victim and suspect (or a chance to gain from the victim’s death), no other alibi for the suspect.
When means, motive and method have been satisfactorily demonstrated to the jury, that can be all that’s required.