Power Broker - S1-E3
Continuity mistake: At the bar, the amount of liqueur in Falcon's snake drink increases between shots.
The Falcon and the Winter Soldier (2021)
1 continuity mistake - chronological order
Starring: Anthony Mackie, Daniel Bruhl, Sebastian Stan, Wyatt Russell
Power Broker - S1-E3
Continuity mistake: At the bar, the amount of liqueur in Falcon's snake drink increases between shots.
Sam: You don't have to trust Redwing. But I'm going to go see if he right. 'Cause I have a feeling they might be part of the Big Three.
Bucky: What Big Three?
Sam: THE Big Three.
Bucky: What Big Three?
Sam: Androids, aliens, and wizards.
Bucky: That's not a thing.
Sam: That's definitely a thing.
Bucky: No, it's not.
Sam: Every tine we fight, we fight one of the three.
Bucky: So who are you fighting now? Gandalf?
Sam: How do you know about Gandalf?
Bucky: I read The Hobbit...in 1937 when it first came out.
Episode #1.6 - S1-E6
Trivia: There's a very important scene with Sharon Carter after the end credits.
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: Aside from visual aesthetics and it being nice to see the metal arm. In world/practically I think (and I believe this is the reason why he tends to not have a left sleeve in most action we've seen him in prior films) it's also to assist with mobility during fights. The metal arm likely behaves a little differently than a human/bone and flesh arm does - and restricting that in a sleeve during a fight (especially in a thicker jacket sleeve) might mess with his reflexes. So removing the sleeve probably assists with the mobility of that arm.
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.