Identify the subject, verb, compliment and prepositional phrase

1. Authorized dealers must give purchasers permit tags as proof of legal sale.
2. What a thorny problem cactus rustling has become.
3. Many work at night and sometimes use permit tags over and over.
4. If you purchase a large cactus, always examine it for bruises.
4. A legally harvested cactus should not show any damage.

1. dealers- subject , must give-verb, purchasers-comp, as proof, of legal sale-pp
2. cactus-subject, has become-verb, problem-comp
3. many- subject, work-verb, at night-pp, over and over-pp
4. cactus-subject, should not-verb, show-comp,

In grammar, it's complement, not compliment. A complement completes the subject (or is the same as the subject).

http://www.chompchomp.com/terms/subjectcomplement.htm

And apparently your text or teacher doesn't want you to identify direct and/or indirect objects, right? They are not to be confused with complements. Read carefully -- http://www.chompchomp.com/terms/ <~~BOOKMARK THIS!
Study the sections on Direct Object and Indirect Object.

1. No
2. Almost
3. No
4. No and no

I'll do one for you:
cactus - subject
should show - verb
damage - direct object

1. dealers- subject , must give-verb, tags-comp , as proof, of legal sale-pp

2. cactus-subject, has become-verb, thorny-comp
3. many- subject, work-verb, at night-pp, over and over-pp
4. always-subject, examine-verb, cactus-comp, for bruises-pp

1. Authorized dealers must give purchasers permit tags as proof of legal sale.

You're almost right: "tags" is a direct object, not a complement.

2. What a thorny problem cactus rustling has become.
"cactus" is not the subject. What is the actual subject of "has become"? You have the verb correct here, but the other two items are incorrect.

3. Many work at night and sometimes use permit tags over and over.
There is an additional verb here. Do you see it? There's also a direct object, but no complement. "over and over" is not a prepositional phrase.

4. If you purchase a large cactus, always examine it for bruises.
This sentence has two clauses in it. I take it you are supposed to concentrate only on the main clause and ignore the dependent clause. If so, you have the verb and the prep phrase identified correctly, but not the subject. For this subject, you need to read up on what a subject can be in an imperative sentence (command).

1. PURCHASERS-COMP

2. Rustling-subject problem-comp
3. use-verb

1. There is no complement in this sentence. There's a direct object (tags) and an indirect object (purchasers), but no complement. To have a complement, the verb needs to be a linking verb, not an action verb. The verb here (give) is an action verb; it's something we can DO!

2. Yes, both are now correct.

3. Yes, "use" is the second verb.

4. YOU-SUBJECT

www.cws.illinois.edu/workshop/writers/complements/

Thank you Mrs. Sue

Be sure you don't confuse direct objects and indirect objects with object complements. Be sure to carefully read the website that Ms. Sue gave you.

bruises-pp

5. you- subject, purchase-verb, cactus-comp, it-pp, for bruises-pp
6. cactus-subject, should not-verb, show-comp, any damage-pp