It is one of our fundamental axioms that God always tells us the truth; which means for many of us that the words he has written in his bible are accurate and true.
Surprisingly, yes. There is an observable proof; just ask yourself. What are the qualities of God? Unless you simply don't speak English or didn't grow up in a Judeo-Christian culture, you innately know. Either your parents told you, or you saw it on TV.
The basic idea is that God is an ultimate moral authority, that he is benevolent; and that he is all-powerful. Therefore, you could look at proofs in the following general areas.
a) If God wanted something to be true, even the meaning of the word 'true' itself, he can make that happen by his will because he is all-powerful.
b) God is perfect so he would not have any of the common needs to lie; deception for self-preservation or personal gain, and so forth
c) If God did lie, then there would be no meaning to any of his word. Therefore the fact we believe in God, that we have his word (bible), and that we have seen his majesty in history and his prophecies unfold in the past and present, suggests that God is telling the truth.
d) We know innately that God tells the truth and he loves us, because we were made in his image. Jesus touches on a similar theme in John 10:
…the sheep follow him because they know his voice. They will not follow a stranger, but they will run from him because they do not know the voice of strangers.”John 10:4-5
Whatever reason you have, for believing it, there's also always the ontological argument.
In any case, we hold this as a self-evident fact: God's word is always true.
The form of this which you can use in Missionary debate is the weak proof that if Jesus (or Paul, etc.) never lied, then he must be from God. This is a weak form of proof because I could always say that George Washington never lied and he wasn't the Messiah. So, not lying is not a strong proof of messiahship.
But you can also make the strong statement that if someone in fact did lie, they would not be from God. For an example of this technique used to start and prooftext a debate please see Did Paul Lie.