prüfen
to check
To check or examine something in order to determine whether it is correct, safe, or functioning properly.
Bevor wir losfahren, muss ich noch den Reifendruck prüfen.
Before we set off, I still need to check the tire pressure.

to check / test
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to check
To check or examine something in order to determine whether it is correct, safe, or functioning properly.
Bevor wir losfahren, muss ich noch den Reifendruck prüfen.
Before we set off, I still need to check the tire pressure.

The verb 'prüfen' (to check / test) is the basis for the noun 'die Prüfung', which means 'test', 'exam', or 'inspection'. The person who checks or tests also has a name: 'der Prüfer' or 'die Prüferin' (examiner / inspector).
Die Studenten bereiten sich auf die mündliche Prüfung vor.
The students are preparing for the oral exam.
When you examine something for a specific quality or feature, you use the preposition 'auf' followed by the accusative. The structure is: 'etwas auf etwas prüfen' (to check something for something).
Die Werkstatt prüft das Auto auf mögliche Schäden.
The garage checks the car for possible damage.
When a person's knowledge in a particular subject is tested, you use the preposition 'in' with the dative. The structure is: 'jemanden in einem Fach prüfen' (to test someone in a subject).
Nächste Woche prüft sie uns in Geschichte.
Next week she will test us in history.
In the meaning of 'überdenken' (to rethink) or 'abwägen' (to weigh up), 'prüfen' (to consider) is very often used with a subordinate clause introduced by 'ob' (whether). This expresses an open possibility.
Die Firma prüft, ob eine Preiserhöhung notwendig ist.
The company is considering whether a price increase is necessary.
In the meaning of 'auf die Probe stellen' (to put to the test), 'prüfen' (to test) is often used in the passive. The subject is then the person experiencing the challenge.
Er wurde durch diese schwere Zeit hart geprüft.
He was severely tested by this difficult time.
to convince
'sich überzeugen von' and 'prüfen' are connected when you want to know whether something is true, safe, or works. 'prüfen' describes the check or examination itself. You look closely, test something, or consider a possibility. 'sich überzeugen von' describes more the goal or result: you make sure of something. Often you check something in order to satisfy yourself of it. 'prüfen' sounds more factual and fits well with technology, documents, or plans. 'sich überzeugen' fits when the person wants to be certain in the end.

Am Montag prüft Frau Richter im Büro die neuen Verträge.
On Monday, Ms. Richter checks the new contracts in the office.
Vor der Abfahrt überzeugte sich Moritz davon, dass alle Fenster geschlossen waren.
Before leaving, Moritz made sure that all the windows were closed.