This is a straightforward challenge, though sometimes tricky to pull off in practice. The floor of the arena in Deck 13.5 is covered in a red digital field that resembles lava, and any Vault Hunter stepping on the floor will incur burn damage. The challenge will activate after EOS's death if the Vault Hunter has not taken damage from the floor during the battle.
This is made slightly more complicated due to knockback effects from enemies/explosions, and by the raised-platform nature of the arena. Vault Hunters jumping from one platform to the next will need to take care not to get hit too hard enroute (or worse, get Crippled in midair), and to keep an eye on their trajectory.