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Most often, the example case given is something like, "there is no gold in Romania" (how I first heard it)

You cannot prove a negative, because you don't have perfect knowledge – you could survey every Romanian in Romania, determine they have no gold, and make the statement – but you have to assume:

• they are not lying
• no one acquired/accumulated gold after you asked
• there isn't some hidden reserve no one knows about

Another example is, "there is no intelligent life in the universe (except on Earth)"

How could you possibly "prove" that statement? Do you know the status of every possible form of "life" that could possibly exist? Do you know the current status of every planet/asteroid/meteor/star/whathaveyounot that exists?

The answer to both of those is clearly "no" – because you (and I) are finite creatures bound by time

If you were a transcendent being (eg the Judeo-Christian view of God), you could definitively state (without "proof", because you were omniscient) something and "know" it was true

But you're not God (or, if you are, you and me need to chat!)

All you need is to introduce (or prove) the possibility of imperfect knowledge, and you can "prove that you can't prove something"

FWIW – this is how the American legal system is supposed to work: when an accused person is convicted of a crime, it is supposed to be "beyond reasonable doubt". The prosecution is not required to "prove" guilt. The defense is not required to "prove" innocence. I once sat on a murder trial jury where we wanted the accused to be "not guilty". We wanted someone else to be guilty. But the defense was unable to introduce "reasonable doubt" to the case. Was the defendant proved to be guilty? No. Not directly – the prosecution didn't have the defendant on video, no "full" eye-witnesses, no "perfect" singular evidence was presented. But all of the circumstantial evidence added up to, "the defendant is guilty".