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Back in the early 2000s I was given a VHS that had a compilation of anime. One particular I remember very fondly.

A vampire and his apprentice fighting other supernatural creatures. At the end of the episode, a human pleads with the vampire to turn him because he doesn't feel like he belongs.

The vampire refuses and in the human's attempt to convince the vampire, he falls to his death from atop a building.

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This sounds like Vampire Princess Miyu:

A vampire and his apprentice fighting other supernatural creatures.

Miyu is a vampire. She has a companion called Larva, although they don't have a master/apprentice relationship. They do fight a variety of supernatural creatures together.

At the end of the episode, a human pleads with the vampire to turn him because he doesn't feel like he belongs.

The vampire refuses and in the human's attempt to convince the vampire, he falls to his death from atop a building.

This is more or less the plot of the first episode. Wikipedia's summary of it:

Miyu Yamano is currently a high school transfer student. A boy named Nobuo Machiyama is aware that she is a vampire that mysteriously killed a girl named Yoko the previous night at a telephone booth. Later on, he visits his teacher Maiko Yanagihara to tell her that Miyu might be targeting her as the next victim. Yanagihara was spotted by a female student named Satsuki Morishita when she had shoplifted a box of chocolates at a store, but she tries to deny that she is in danger of getting caught. To avoid this, she summons her pet chameleon, which transforms in a shinma named Ga-Ryu, and attacks Satsuki head-on. Miyu appears and banishes Ga-Ryu into the darkness as a consequence. Miyu then puts Yanagihara to an eternal sleep, which has her tending to her garden full of flowers. Machiyama accidentally falls to his death off of the school's rooftop after failing to convince Miyu to allow him to kill shinmas alongside her.

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