Character mistake: Bucky says he read The Hobbit in 1937 when it first came out. But in 1937 he was still living in New York, and The Hobbit was only available in the UK at that time, with fairly limited numbers printed. It wasn't available in America until 1938.

The Falcon and the Winter Soldier (2021)
1 mistake
Starring: Anthony Mackie, Daniel Bruhl, Sebastian Stan, Wyatt Russell
Bucky: I got a vibranium arm. I can take them.
Sam: And I can fly, who gives a shit? Wait.
Power Broker - S1-E3
Trivia: When Dovich and Karli are speaking in the street in Vilnius, Lithuania, visible in the background is a gate being guarded by several men, and on the gate is an encircled X, very reminiscent of the X-Men logo. The fictional nation of Madripoor seen in this episode is also closely associated with the X-Men in the comics. (00:38:57)
Question: Why does Bucky rip his jacket sleeve off before jumping out of the plane? Just for the sake of an audience reveal to demonstrate his artificial arm?
Answer: That's the impression I got. Plus, he was expecting to go into battle, so perhaps he thought it would make him look more intimidating.
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Chosen answer: The sleeve probably would have been ripped to shreds in battle anyway. He was just saving time by ripping it himself. Also, gunfighters would take off their coats in a duel or an arm wrestler rolling up his sleeve.