Dette beviset kan bevise at påstanden er feil.

In Norwegian, the is usually attached to the end of the noun.

• et bevis = a proof
• beviset = the proof

So in dette beviset, the -et ending is what makes bevis definite.

Also, with demonstratives like dette (this), Norwegian normally still keeps the noun in the definite form:

• dette beviset = this proof

That is different from English, where you do not say this the proof.

Because bevis is a neuter noun.

The indefinite form is:

• et bevis = a proof

For singular demonstratives, Norwegian uses:

• denne with common-gender nouns
• dette with neuter nouns

So:

• dette beviset = this proof
• but denne påstanden = this claim, because påstand is a common-gender noun: en påstand

Because they are different forms built from the same root:

• beviset = the noun the proof
• bevise = the verb to prove

So the sentence contains both:

• Dette beviset = This proof
• kan bevise = can prove

English has something similar with related words like proof and prove, but in Norwegian the forms look even closer.

Because kan is a modal verb, and modal verbs are followed by the bare infinitive in Norwegian.

So you say:

• kan bevise
• må gå
• skal skrive
• vil lære

not:

• kan å bevise

In other words, after kan, Norwegian does not use å.

Here kan means something like can, is able to, or is capable of.

So Dette beviset kan bevise at ... means the proof is sufficient or capable of proving that the claim is wrong.

A slightly stronger version would be:

• Dette beviset beviser at påstanden er feil.

That sounds more like This proof proves that the claim is wrong.

With kan, the sentence leaves a little more room for ability or potential.

At means that and introduces a subordinate clause.

• bevise at ... = prove that ...

So the sentence is structured like this:

• Dette beviset = subject
• kan bevise = verb phrase
• at påstanden er feil = that the claim is wrong

English sometimes drops that, but Norwegian often keeps at in sentences like this.

Because it refers to a specific claim, not just any claim.

• en påstand = a claim
• påstanden = the claim

So påstanden er feil means the claim is wrong.

This suggests that the claim is already known from the context.

It can, but here feil is being used as an adjective-like word meaning wrong or incorrect.

So:

• påstanden er feil = the claim is wrong

Compare:

• Det er en feil = There is an error / a mistake

In your sentence, feil is not a noun. It is describing påstanden after er.

After at, Norwegian uses subordinate clause word order, which here is:

• påstanden = subject
• er = verb
• feil = complement

So:

• at påstanden er feil = that the claim is wrong

This is a normal pattern in Norwegian subordinate clauses.

The full sentence is:

• main clause: Dette beviset kan bevise ...
• subordinate clause: at påstanden er feil

It is grammatical and understandable, but it can sound a little repetitive because bevis and bevise are so closely related.

A Norwegian speaker might also say:

• Dette beviset viser at påstanden er feil.
  This evidence shows that the claim is wrong.

or:

• Dette beviser at påstanden er feil.
  This proves that the claim is wrong.

So your original sentence is correct, but stylistically it is a bit heavy because of the repeated bevis- root.