Becky: Yes.
For myself, I don't find this convincing. Here's why. "Scripture is inspired because it claims to be inspired" is self-referential. If I say to you, "Becky, you can trust me, because I always tell the truth". If you question me with, "But angelo, how can I know that it is true that you always tell the truth?" then I respond, "because I said so, and I always tell the truth". The argument is circular.
Here are what I consider two non-circular arguments for believing that scripture is inspired:
1. God has independently revealed to me that these writings are inspired. It is not circular because the evidence for the inspiration comes from outside the writings in question.
2. God revealed the authenticity and inspired nature of these writings to others, who have passed their testimony onto me.
How do you know the OT is divinely inspired?
#2, above. The truth of the Hebrew scriptures is grounded in the testimony of the people of Israel who witnessed the events at Sinai. These original witnesses passed their testimony onto the next generation, who passed it down to each subsequent generation until it reached me.
Similarly, you could argue the inspiration of the Christian scriptures on the basis of the original witnesses of the life of Jesus. They wrote the gospels, and passed them onto the next generation with the testimony that the events recorded therein were true. Each generation then passed them down with this testimony to their authenticity, until the present day. Thus you have an independent, oral tradition testifying to the authenticity of the gospels.
This is my answer to the question: I believe in God and the Bible because my parents told me so. I trust my parents and if they believed it, and I haven't found any reason to doubt it, I can't see how I can not believe it. It is exactly this intergenerational transmission of revealed truth that so infuriates folks when their children leave "the faith," regardless of what faith it is. It is essentially the essence of not "honoring your mother and father" to take what they have held as true and brought to you as true over millennia and to discard it.
SD