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Prepare for the climb. Learn the “Dragonrend” shout. Use “Dragonrend” to bring Alduin to the ground. Use “Clear Skies” tactically. Avoid Alduin’s front. Use frost magic. Apply poisons to your weapons. Keep the pressure on.
Purchase or craft a number of healing and fire resist potions in any town. You may want to consider recruiter a follower to help you as well.  A solid melee fighter, like Lydia or Mjoll are good choices to tank Alduin when he lands. Ilia is also a good choice for her proficiency in frost magic. There will be a few Ice Wraiths on the climb, so a couple of frost resist potions may be a good idea as well. If you have training in Restoration magic, you can use spells to substitute for healing or resist potions. When you reach the top, you will speak to Paarthurnax and watch a lengthy cutscene to learn the shout before the fight with Alduin begins. Paarthurnax will aid you in your fight against Alduin and act as a distraction, but you will have to deal the bulk of the damage. Tap and hold the shout button to charge it. Follow Alduin’s flight with the camera as you charge so that you do not miss. If you miss you will have to wait while your shout recharges.  Equip an Amulet of Talos if you need the shout cooldown to be shortened. Although Paarthurnax will spar with Alduin in the air, Alduin cannot be damaged until you ground him with Dragonrend. A single tap of the shout button will do a single-charge version of the shout that can still effectively stun Alduin after he is already grounded. You can use this to help preventing him from becoming airborne again. Alduin has his own shout that causes meteors to rain from the sky. When this happens, you can use Clear Skies to cancel it.  You will have learned this shout from The Greybeards before being allowed to ascend the mountain, so you cannot have missed it. If your shout is on cooldown when this happens, then it is a good time to use a fire resist potion. Remember, using Clear Skies means your shout will go on cooldown and you will have to wait for it to recharge before you can use Dragonrend again. Once on the ground, Alduin uses a powerful fire breath attack that does lots of damage to everyone in front of him and a strong tail whips from the back. Try to attack from the sides regardless of range or melee.  Eventually he will turn to bite you, causing you to face his front again, so keep strafing to the sides. Fire resist potions can help if you want to challenge him from the front, but best to avoid it altogether. If you have training in destruction magic, spells like ice spike and ice storm can do big damage to Alduin since he has a slight weakness to frost.  Ice storm can damage your follower, so use it with caution. If you have it, the Marked For Death shout will greatly increase your damage with magic, but timing it between uses of Dragonrend and Clear Skies can be difficult. Lingering poisons are just as effective against Alduin as any other non-undead creature. These can be crafted or acquired from an alchemy shop in any town. Using poisons if very effective when using a bow from afar while a follower tanks Alduin’s hits at melee range. Keep Alduin on the ground with Dragonrend, use health potions when you get low, and before long Alduin should succumb to the damage and retreat.