to instruct / lecture
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to prove wrong
To show or prove to someone that their previous assumption or opinion was wrong.
Ich dachte, die Aufgabe sei unmöglich, aber er hat mich eines Besseren belehrt.
I thought the task was impossible, but he proved me wrong.
The prefix 'be-' is inseparable. It remains firmly attached to the verb stem 'lehren' (to teach) in all tenses and sentence constructions.
Er hat mich darüber belehrt, wie man die Maschine richtig bedient.
He instructed me on how to operate the machine correctly.
The expression 'jemanden eines Besseren belehren' (to prove someone wrong) requires the genitive 'eines Besseren'. The structure is fixed and cannot be changed.
Sie glaubte nicht, dass ich es schaffe, aber ich habe sie eines Besseren belehrt.
She didn't believe that I could do it, but I proved her wrong.
When 'belehren' (to lecture/instruct) is used without the addition 'eines Besseren', it often has a negative, slightly arrogant meaning. It implies that someone is giving advice in a pushy or condescending way.
Bitte hör auf, mich ständig zu belehren; ich weiß, was ich tue.
Please stop constantly lecturing me; I know what I'm doing.
The expression 'eines Besseren' signals the neutral meaning of showing someone that they were wrong. Without this addition, the negative, admonishing meaning is usually intended.
Er belehrte mich eines Besseren und zeigte mir meinen Fehler.
He proved me wrong and showed me my mistake.
to explain / declare
'erklären' and 'belehren' can both mean that someone tells another person something so that they understand or learn it. The most important difference is the tone. 'erklären' is neutral and often helpful. 'belehren' often sounds negative because the person is speaking condescendingly or wants to show someone that they have done something wrong. In many everyday situations, 'erklären' is more natural and friendlier. 'belehren' is more appropriate when the tone seems strict, reproachful, or condescending.
Der Kollege hat mich vor allen belehrt, obwohl ich nur eine Frage gestellt habe.
My colleague lectured me in front of everyone, even though I had only asked a question.
Im Laden erklärt mir die Verkäuferin den Unterschied zwischen den beiden Modellen.
In the store, the sales assistant explains the difference between the two models to me.