The phrases sets in, sets out, sets off, and sets up have similar but distinctive idiomatic senses as intransitive verb phrases, which I would express as follows:
sets in: arrives, becomes fixed, or begins to take effect. Example: It will be harder to plow the fields when winter sets in.
sets out: undertakes, or begins to make an effort. Example: Once he sets out to answer a question, he won't rest until he has succeeded.
sets off: departs, Example: Every morning at dawn, she sets off for her job 5 miles away.
sets up: prepares for business or for use (this is more often a transitive verb phrase, but it can be used intransitively in some situations) Example: He brings his goods to the market stall each day and then sets up for business.
The American Heritage Dictionary of the English Language, fifth edition (2011) offers the following more formal definitions for each phrase:
set in: 1. To insert: set in the sleeve of a gown. 2. To begin to happen or be apparent: "Evening was setting in as I as I took the road over Mountain Top" (Charles Siebert).
set out: 1. To begin an earnest attempt; undertake: He set out to understand why the plan had failed. 2. To lay out systematically or graphically: set out a terrace. 3. To display for exhibition or sale. 4. To plant: set out seedlings. 5. To start a journey: She set out at dawn for town.
set off: 1a. To give rise to; cause to occur: set off a chemical reaction. b. To cause to explode: set off a bomb. c. To make suddenly or demonstrably angry: The clerk's indifference finally set me off. 2. To indicate as being different; distinguish: features setting him off from the crowd. 3. To direct attention to by contrast; accentuate: set off a passage with italics. 4. To counterbalance, counteract, or or compensate for: Our dismay at her leaving was set off by our knowing that she was happy. 5. To start on a journey: set off for Europe.
set up: 1. To place in an upright position. 2a. To elevate; raise. b. To raise in authority or power; invest with power: They set up the general as a dictator. c. To put oneself forward as; claim to be: He has set himself up as as an authority on the English language. d. To assemble and erect: set up a new machine. 3. To establish; found: set up a new charity. 4. To cause: They set up howls of protest over new taxes. 5. To establish in business by providing capital, equipment, or other backing. 6. Informal a. To treat (someone) to drinks. b. To pay for (drinks). 7. Informal To stimulate or exhilarate: a victory that really set the team up. 8. To lay plans for: set up a kidnapping. 9. Informal To put (someone else) into a compromising situation by deceit or trickery: Swindlers have set me up. 10. Sports To make a pass to (a teammate), creating a scoring opportunity.
The key to negotiating this mass of definitions is to look for options that involve intransitive verbs, the form needed to complete the original sentence: "The author ____ ____ to prove..." Quickly we can winnow the relevant American Heritage definitions down to sets in definition 2; sets out definitions 1 and 5; sets off definitions 4 and 5; and sets up definition 5 (arguably).
The author doesn't begin to happen or begin to be apparent to prove X, so sets in is no good as the answer. The author doesn't counteract or counterbalance to prove X and doesn't start on a journey to prove X (except in a rather strained figurative sense), so sets off looks pretty unsatisfactory. And the author doesn't set up in business to prove X (except, again, in a strained figurative sense), so we can discard sets up.
That leaves sets out, and the first American Heritage Dictionary definition—"To begin an earnest attempt; undertake"—seems exactly on point: The author begins an earnest attempt or undertakes to prove X. It's true that you could push the figurative meaning "start a journey" (which is no better or worse than the intransitive meaning of set off we considered earlier)—but you don't need to do that. You already have a more suitable definition of set out confirming sets out as the right phrase.