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Before fighting the Elite Four, do the following:

1. Make sure your Pokemon are Level 50+.
2. You have plenty of Full Restores.
3. You have plenty of Revives.

Once all of these have been done, the Elite Four should be challenged.

Jynx is of the Ice/Psychic combo, and Fire moves will become very helpful in taking her down. Lapras, Cloyster, Slowbro, and Dewgong, however, are of the Ice/Water and Water/Psychic combo, and the Water type nulls the Fire type's super effectiveness. Electric moves are better on them. Fighting type moves can work also.

Onix has the type combo of Rock/Ground, and thus giving him great weakness to Water and Leaf Base moves. Hitmonlee, Hitmonchan, and Machamp are weak to Psychic and Flying moves like before. However, any strong physical move (Mega Punch, Thrash, etc) can easily take out the Hitmon's because of their low defense stats.

Golbat, Haunter, Arbok, and both of her Gengar share the same weakness, Psychic type moves. Speedy Psychic Pokemon, or a fast Pokemon that knows such moves, will be needed or the Gengar can quickly decimate your party.

Gyarados is greatly weak to Electric moves, so use them. Both Dragonite and his Aerodactyl are greatly weak to Ice moves. Same with his Dragnair.

Pidgeot is a Flying type, so Electric moves are good to use. Rock moves are also effective. Rhydon is like Onix with the Rock/Ground combo, so Water and Leaf moves will wipe the floor with him.

Alakazam, for a Psychic, has high defense, so a good elemental blast (Flamethrower, Ice Beam, Thunderbolt) will set it in its place.

As for his Fire, Water, and Leaf Pokemon. Water > Fire, Leaf > Water, Fire > Leaf. Simple.