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By definition, a positive real number is a real number greater than zero.  That statement cannot be proved to be right; it cannot be proved to be wrong.  We either reject it and use a different definition, or we accept it and move on.

Consider the statement "$1-1+1-1+... = 0$."

Here is a "proof" that it is right: $1-1+1-1+... = (1-1)+(1-1)+... = 0+0+... = 0$

Here is a "proof" that it is wrong: $1-1+1-1+... = 1-(1-1)-(1-1)-... = 1-0-0-... = 1 \ne 0$

But that is absurd; a statement cannot be right and wrong at the same time.  Either we give up and say that the expression is meaningless, or we can *define* it be equal to some value.  If the statement is a definition, then it cannot be proved to be right and it cannot be proved to be wrong.  We either reject it and use a different definition, or we accept it and move on.