Мне всё-таки удалось убедить брата пересесть: в первом вагоне не было ни одного свободного сиденья.

Because удаться / удалось is used in an impersonal construction in Russian.

Literally, Мне удалось убедить брата is something like:

• It succeeded for me to convince my brother
• or more naturally, I managed to convince my brother

So:

• мне = to me / for me in the dative case
• удалось = it worked out / it was possible / I managed

Russian often uses this kind of structure where English would simply use I as the subject.

In this construction, удалось is the past tense of удаться, and it is usually used in the neuter singular because the sentence is impersonal.

There is no normal subject like я here. Russian treats the whole situation as something like it turned out successfully.

So:

• Мне удалось... = I managed...
• Ему удалось... = He managed...
• Нам удалось... = We managed...

Notice that удалось stays the same even though the person changes; the person is shown by the dative pronoun.

Всё-таки adds the idea of after all, in the end, nevertheless, or despite everything.

In this sentence, it suggests there was probably some difficulty, hesitation, or resistance before the speaker succeeded:

• Мне всё-таки удалось... = I did manage, after all...

It often gives a mild emotional nuance: the result happened, maybe not easily.

Compare:

• Мне удалось убедить брата = I managed to convince my brother
• Мне всё-таки удалось убедить брата = I did manage to convince my brother in the end / after all

Убедить means to convince or to persuade successfully.

It is the perfective form, so it focuses on the result: the brother ended up being convinced.

Compare:

• убеждать = imperfective, the process of trying to convince
• убедить = perfective, successfully convince

Here the speaker is talking about a completed achievement, so убедить is the natural choice.

Because убедить takes a direct object, and brother is that object: convince whom?

So we need the accusative case:

• брат = nominative
• брата = accusative

Since брат is a masculine animate noun, its accusative singular looks like the genitive singular:

• Я вижу брата
• Я убедил брата

This is a very common pattern in Russian.

This is a common Russian pattern:

• убедить кого? сделать что?
• to convince someone to do something

So:

• убедить = to convince
• брата = brother, the person being convinced
• пересесть = to move / change seats

Altogether:

• убедить брата пересесть = to convince my brother to move to another seat / change seats

Russian often uses an infinitive after verbs like просить, заставить, убедить, позволить, etc.

Пересесть literally means to move from one seat to another or to change seats.

It comes from:

• сесть = to sit down
• пересесть = to sit somewhere else / move over / switch seats

In other contexts, пересесть can also mean:

• to transfer from one train/bus/etc. to another

But here, because the sentence talks about free seats, it clearly means change seats / move to a different place.

Because the speaker is talking about a single completed action: the brother was convinced to make one change of seat.

Perfective infinitives are very common after verbs that describe achieving a result:

• удалось убедить брата пересесть

This means the action is viewed as a whole, completed event.

If you used an imperfective infinitive here, it would usually change the nuance and sound less natural in this context.

The colon introduces the reason or explanation for what comes before it.

So the structure is:

• I managed to convince my brother to change seats: there wasn’t a single free seat in the first carriage.

The second part explains why changing seats was necessary or why the speaker was able to convince him.

In English, you might also use a colon, dash, or words like because depending on style.

Because в with location takes the prepositional case.

Here it means in the first carriage/car:

• в = in
• первом = prepositional singular of первый
• вагоне = prepositional singular of вагон

So:

• в первом вагоне = in the first carriage

If motion into something were meant, you would often expect a different case pattern, but here it is clearly location.

Вагон usually means a railway car / carriage. Depending on context, English might translate it as:

• carriage
• car
• railway car
• sometimes train car

So в первом вагоне means in the first carriage / in the first car of the train.

Не было comes from не было = there was not / there were not.

This is another very common impersonal existence construction in Russian:

• В комнате не было стола. = There was no table in the room.
• В вагоне не было свободных мест. = There were no free seats in the carriage.

Russian often uses было / не было in a neuter singular form for statements of existence or absence.

So:

• в первом вагоне не было... = in the first carriage there was not...

Natural English: there wasn’t...

Russian commonly uses double negation:

• не on the verb
• plus ни in the noun phrase

So:

• не было ни одного... = there wasn’t a single...

This is normal and required in standard Russian.

Compare:

• Я никого не видел. = I didn’t see anyone.
• Там не было ни одного места. = There wasn’t a single seat there.

In English, double negation usually changes meaning or sounds nonstandard, but in Russian it is a basic grammatical feature.

Because after ни одного the noun goes in the genitive singular.

The basic noun is:

• сиденье = seat

Its genitive singular is:

• сиденья

So:

• ни одного сиденья = not a single seat

This is exactly what Russian grammar requires after один / одного in this negative-existential pattern.

Because adjectives must agree with the noun they describe.

Here the noun is сиденья, which is:

• genitive singular
• neuter

So the adjective свободный becomes:

• свободного

Thus:

• ни одного свободного сиденья = not a single free seat

Both свободного and сиденья are in the genitive singular.

Yes, but it would mean something slightly different in structure.

Two natural options are:

• не было ни одного свободного сиденья = there wasn’t a single free seat
• не было свободных сидений = there were no free seats

The version in your sentence is more emphatic because ни одного highlights not even one.

The word order is natural and information-driven.

• Мне всё-таки удалось puts focus first on the speaker’s success after difficulty.
• убедить брата пересесть gives the completed result.
• after the colon, в первом вагоне sets the scene first
• не было ни одного свободного сиденья gives the important new information

Russian word order is flexible, but this order sounds very normal and clear. It presents the sentence almost like:

1. I did manage in the end
2. to convince my brother to change seats
3. because in the first carriage
4. there wasn’t a single free seat

They are related, but not always identical.

• сиденье = the physical seat
• место = place / spot / seat in a broader sense

In transport contexts, both can sometimes be translated as seat, but:

• свободное место often means an available place to sit
• свободное сиденье emphasizes the actual seat itself

In this sentence, свободного сиденья is perfectly natural because the context is specifically about seating in a train carriage.