That’s about as true as saying that a point in space is always an approximation. In a mathematical context, yes, the definition of a point is always subject to greater precision. In the context of describing the location of objects (as well as the degree that a claim is “proven”) does not depend on an infinitesimally small measurements.
I can prove that a hungry tiger is dangerous. But because I can not provide evidence that sustains itself to 10 to the millionth, billionth or trillionth power does not make my judgment that danger exists a “subjective” one. One does not have to thrash in the inability to disprove the one in a trillion-trillion chance that everyone’s eyes are consistently lying to them in order to pronounce something proven.
In this context of human life, it’s provable by overwhelming evidence that tigers are dangerous. Our experiences are similar enough to call that an “objective” claim. But as far as I know, there is no similar overwhelming evidence that “proves” the validity of the Bible.
Of course if you want to preface your claims of proof being subjective as assuming that the most infinitesimally small amount of doubts precludes objectivity, then I won’t argue that it’s true in that narrow mathematical context, removed from the human experience.
You are dodging my point: that all "proof" is apprehended through the senses, which apprehension is and must be a subjective experience. One cannot prove that tigers are dangerous, or even that tigers exist: all one can say with certainty is that one perceives certain sensory inputs that lead one to believe that tigers exist. The issue of whether or not those sensory inputs correspond to reality in any meaningful way is impossible to know with certainty; one might, for example, be merely dreaming of a tiger. At any given instant there is no way to determine with certainty which sensory inputs correspond to reality and which do not; for all we know, everything we see and do is merely part of a dream, or a fantasy, or a hallucination, or a drug trip. All we may know with certainty is that we ourselves exist; we do not see, hear, smell, touch, or taste our own selves — we are ourselves.
Of course, I believe that tigers exist, and that they can be dangerous. But I cannot say that this is certainly true; I can only say that I accept its truth as a working assumption — in other words, as a matter of faith. The same goes for the existence of atoms, energy, the internet, elfman2, and the rest of the universe. One either makes a leap of faith and believes that Reality exists outside of one's own skull, or one becomes a solipsist.
Only that which is directly experienced can be known with certainty to exist. Therefore, sensory perception can not be the sine qua non of reality. Imagine a person placed in a sensory deprivation tank, with all sensory inputs absent — unable to feel, see, hear, touch, taste, or smell anything at all. Furthermore, let us assume this imaginary person has been dosed with a powerful anesthetic drug that deadens his sense of his own body positioning while allowing him to remain conscious. What could such a person know to exist without sensory "proof"?
Answer: he could know his own existence. He could "hear himself think". Despite the utter lack of evidence, he could know with 100% certainty that he himself existed. This is what I mean when I say that "proof" is an illusion: since all "proof" is based upon sensory data, which is inherently subjective, then the notion of "objective proof" or "objective existence" is meaningless. The senses can be fooled — but one cannot doubt that which one experiences directly, without use of the senses, i.e. one's own existence.
And so we see that your notion of "proof" makes no sense, because it is founded upon the senses. As Socrates pointed out, the truly wise man is he who knows that he knows nothing— the man who heeds the words of the Delphic Oracle and knows only that which can be known: "Know Thyself".