what convinces the Otises that the ghost does exist?
The Otises are convinced that there is no ghost in the Canterville house. The stories are not enough to convince them, and the weird noises don't sell them on the existence of the ghost either.
The family begins to get suspicious and consider that the ghost might be real after they fail to remove the blood stain with the special stain remover. The Otises tried three times to clean the stain with Pinkerton's Champion Stain Remover and Paragon Detergent, but the stain kept coming back by the next morning.
Shortly after that though, all doubt was removed about the ghost.
"The following night, all doubts about the existence of the ghost were finally removed forever."
The reason all doubts were removed was because Mr. Otis was awakened at 1 in the morning by the sound of clanking metal. He went out to find out what was making the noise and came face to face with the ghost -- red eyes and all.
"There, right in front of him, stood the ghost - his eyes were as red as burning coals; long grey hair fell over his shoulders and from his wrists and ankles hung heavy chains."
The best part of the entire encounter though is Mr. Otis telling the ghost how he really should oil those chains, so that they do not make so much racket.
'My dear Sir,' said Mr Otis, 'you must oil those chains. It's impossible to sleep with such a noise going on outside the bedrooms. I have therefore brought you this bottle of lubricator, and I will be happy to supply you with more if you require it.' With these words Mr Otis laid the bottle down, closed his door and went back to bed."