What kind of proof do they desire? Perhaps they desire a scientific proof, such as a mathematical derivation. But mathematical proofs are regularly found to be flawed and are based only on the limited logic of men. Perhaps more people would be convinced by empirical evidence. But this too is subject to interpretation and argument. Even some of the most "proven" scientific theories are regularly questioned, reconsidered, and reworked. In spite of all the evidence, scientific theories are just that--they are theories and are limited by the intellect of the men and women who create and subscribe to them. All "proof" is simply evidence that men in their limited minds consider very strong. These days, scientific knowledge and human logic seem to be considered very strong "proof."
Heavenly Father does offer proof of His words. But His proof does not come in the form of mathematical derivations or scientific evaluations. God is not a scientist. He already knows everything. He could personally tell a scientist or any individual what He knows, but would the person be convinced, or would they endlessly question the knowledge they receive? He could personally appear to someone, but would the memory of that experience be enough to convince them of its reality, or would alternate explanations be considered? He could move a mountain or bring someone back from the dead, but neither of these things in and of themselves are enough to completely convince someone. Mountains could be moved by extreme plate tectonics; people rising from the dead were perhaps not completely dead anyway, or perhaps a live person was subtly switched with the dead person, or perhaps there is simply some unknown alternate explanation for what we see. God could miraculously heal the sick, calm ocean storms, or change water into wine before our very eyes, yet many would remain unconvinced. He demonstrated this during His sojourn on earth.
What then is Heavenly Father's proof? What possible proof can He give us that we will believe? Put simply, His proof is the realization of the promises He makes, together with a heavenly assurance. He tells us plainly that we will know whether what He says is true "If any man will do His will, he shall know" whether His teachings are true or made up. In order to see whether Newton's Laws would work, we had to try them out. Similarly, if we do what God asks of us, we will see the promised blessings, however unlikely those blessings may seem at the time. Also, by submitting to God's will, we build a relationship with Him that allows Him to communicate with us heart-to-heart, and this new means of communication allows us to receive a direct injection of knowledge--we can feel something of what He feels, and experience a reflection of what He knows. This personal connection with God Himself is more convincing than any lecture by an expert in theoretical physics. It is stronger proof than decades of empirical evaluation. To look into the heart and mind of One who knows all things is a priceless venture, and it does not require an academic degree. What it does require is a willingness to try--faith to live as God would have me live. As I give my heart to Him, He opens His heart to me.
Countless millions have experienced personal proof of His promises. Though the proof comes in a different form than some desire or expect, it is stronger and more reliable than any proof of a worldly nature.