(Plural) + "are" + (noun) – is this usage correct?

Question

I read the following sentence:

Mistakes are proof that you are trying.

It is correct to use "are" followed by a noun?

Why didn't the author say "mistakes proof" instead of "mistakes are proof?"

Answer 1

You can use a mass noun (which "proof" is) or a plural here. For example:

Mistakes are proof that you are trying.
Proof is a mass noun.

These are opportunities that you're squandering.
Opportunities is a plural.

You can't use a singular because you are referring to multiple (i.e. a plurality of) mistakes.

You can't say "Mistakes proof" because "proof" isn't a verb that can be used in this way. You could, however, say:

Mistakes prove that you are trying.

Answer 2

I will try to explain it in the way I understand this. The thing is that this verb "are" it is not bad in that sentence, why?

"Mistakes are proof that you're trying". "Are" emphasizes the word "Mistakes", I mean, it is like a general definition in that sentence. On the other hand, "Mistakes proof that you're trying" it is the same but more directly, without defining anything, just hit the point. You have to pay attention on the verb "proof", try to change that sentence with another word instead of proof and you'll see the difference on the context.

"Values proof that you're a respectful person". "Values are proof that you're a respectful person".