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¤ What is this section?
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Have two Pokemon around Level 10 or so because Mew is at Level 7. You're capturing it, not killing it. You will also need a Pokemon that knows Fly.

Return to Celadon City and head for the Underground Path, East Saffron Gate. Take one step outside and then save. There is always the chance you could mess up. Move your character down and open the start menu simutaniously. Select the Pokemon that knows Fly and head to Cerulean City.

Now, head to Bill's house and fight this trainer:
>

Defeat him and fly to Lavender Town. Use the Poke-Center if you need to. Now head towards the underground path. When you get on the narrow strech of road, a message will pop up that reads "I'm better at school than this!" and you will engage battle with a Level 7 Mew.

Mew is not really a threat. It only knows Pound and can not do much else. Weaken it to half its health and start throwing Great Balls until you catch it. If you manage to catch it, great job!