I do not believe there is any ONE arguement to be made for inerrancy of the scripture. It is a culmination of many things put together from which we who believe in the inspiration and inerrancy of the scriptures derive our answer. I can only suggest reading some books by folks who hold this position, as I do not think it possible to answer that in a few short sentences.
I will proclaim the goodnews of Christ, I will not try to argue anyone into believing it. My duty is to tell the goodnews, the Holy Spirit's duty is to convict one of their need for it. Once you hear it, you may accept or refect it, if you have problems with it, research it on your own, but I won't argue anyone into the kingdom of God, I don't know that that is possible, although there are some folks who are better qualified than myself that could probably do a great job of it.
I will say, if someone even has the tiniest inclination or interest, that it may be true, then, as Jesus said, seek and ye shall find, knock and it shall be opened unto you. If one truly wants to find out if it is true, that person must put forth some effort of his own; draw nigh to God and he will draw nigh unto you. Nobody can do it for you, it is a personal endeavor, but it has great rewards for those who would seek out of a pure heart. PEACE!