After beating cerulean cities gym head down from cerulean and enter the underground path go through then go down and there is vermillion.
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It's in Cerulean City. You get the Bike Voucher from a guy in Vermillion City,
Vermillion City underwater just kidding LT. Surge gives you it after beating him.
Go to Vermillion city(or whatever its called).Go to the fan club.Go to one of the people and he gives you it.
In Vermillion City there is a Pokemon Fan Club if you listen to the Chairman of the Fanclub talk about his Rapidash, he will apologise to you and give you the Bicycle Voucher which you can redeem for a bike in Cerulean City.
Vermillion City is located below Saffron City. You can get there only after you meet Bill, the Poke'maniac.