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One of the pregen fighters in the Starter Set has proficiency in simple & martial weapons, giving a +2 bonus, but also has the Archery fighting style, giving a +2 bonus.

I don't see these added together. Why bother adding 'Archery' if a +2 bonus is already added in proficiencies; does the Archery option offer anything later on?

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Both of those modifiers do add to the attack roll; the combined modifier is then relevant ability mod (DEX, +3) + proficiency bonus (+2) + Archery bonus (+2), yielding the +7 you see in the "Attacks & Spellcasting" section of the pregen character sheet (available from Wizards here).

pregen ftr (version 2) from starter set

(Note that the +4 attack bonus for greatsword is just the relevant ability mod (STR, +2) + proficiency bonus (+2).)

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A character's Proficiency Bonus and the Fighter's Fighting Style class feature do stack.

Proficiency with something in 5e allows you to add your Proficiency Bonus to the relevant rolls. This is a dynamic bonus that scales with character level (not class level) and caps at +6 at level 20.

Archery is a static +2 bonus to attack rolls made with ranged weapons granted when you choose this feature. Also note that Fighting Style is not a class feature unique to the Fighter.

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