De kan sætte Dem ned i venteværelset.

They are capitalized because they are the formal second-person pronouns in Danish.

• De = formal you (subject form)
• Dem = formal you (object form)

Capitalizing them helps distinguish them from:

• de = they
• dem = them

So the capital letters are not random—they show that the sentence is addressing someone politely and formally.

De/Dem is the formal, polite way to say you, while du/dig is the normal informal singular you.

Compare:

• Du kan sætte dig ned. = informal, everyday
• De kan sætte Dem ned. = formal, polite, somewhat old-fashioned or very respectful

Today, Danish usually prefers du in most situations, much more than English uses formal language. So De/Dem can sound very polite, distant, traditional, or sometimes old-fashioned.

Because Dem is the object form.

In the sentence:

• De is the subject: the person doing the action
• Dem is the object after sætte

This is similar to the English difference between I/me, he/him, or older English thou/thee.

So:

• De kan ... = You can ...
• ... sætte Dem ned = ... sit yourself down

Sætte Dem ned is the formal version of the common Danish expression sætte sig ned, which means sit down.

Literally, it is something like:

• sætte = set/place
• Dem = yourself / you
• ned = down

So the structure is basically set yourself down → sit down.

This is a very common kind of expression in Danish. The reflexive pronoun changes depending on the person:

• Jeg sætter mig ned = I sit down
• Du sætter dig ned = you sit down
• De sætter Dem ned = you sit down (formal)

Because sig is used for the third person, not for De.

In modern Danish grammar:

• mig = myself / me
• dig = yourself / you
• sig = himself / herself / itself / themselves
• Dem = yourself / you (formal)

So sig would not match De here. Since the sentence directly addresses the listener formally, Dem is the correct reflexive/object form.

Grammatically, kan is the modal verb can. But in this kind of sentence, it often sounds less like real ability and more like a polite instruction, permission, or invitation.

So De kan sætte Dem ned i venteværelset can feel like:

• You may sit down in the waiting room
• You can have a seat in the waiting room
• Please sit down in the waiting room

This is common with kan in Danish: it can soften what is basically an instruction.

Because Danish modal verbs normally take a bare infinitive, just like English.

So:

• kan sætte
• not kan at sætte

This is similar to English:

• can sit
• not can to sit

Common Danish modal verbs that work this way include:

• kan = can
• skal = shall/must
• vil = will/want to
• må = may/must
• bør = ought to

This is normal Danish main-clause word order.

The finite verb comes in second position:

• De = first element
• kan = finite verb in second position
• sætte = infinitive later in the sentence

So the pattern is:

Subject + finite verb + infinitive + object + particle/adverbial + place

That is very typical in Danish main clauses.

If another element came first, the verb would still stay in second position. For example:

• I venteværelset kan De sætte Dem ned.

That is one of the most important word-order patterns in Danish.

Venteværelset is a compound noun:

• vente = waiting
• værelse = room

So venteværelse means waiting room.

The ending -et is the definite article attached to the noun, so:

• et venteværelse = a waiting room
• venteværelset = the waiting room

Danish usually adds the definite article to the end of the noun instead of putting a separate word like the in front.

Because i is the normal preposition for being in a room or enclosed space.

So:

• i venteværelset = in the waiting room

Danish often uses i where English uses in:

• i huset = in the house
• i bilen = in the car
• i venteværelset = in the waiting room

Using på would usually suggest a different spatial relationship and would not be the normal choice here.

Yes, but it sounds formal.

A modern everyday version would more often be:

• Du kan sætte dig ned i venteværelset.

In many real situations, Danes might also say something slightly more idiomatic, such as:

• Du kan tage plads i venteværelset.
• De kan tage plads i venteværelset.

Tage plads means something like take a seat and is very natural in polite service situations.

So the original sentence is correct, but the use of De/Dem makes it more formal than what you usually hear in everyday Danish.