Others have commented on the manuscript evidence and the fact that there were 7000 other faithful Jews who had not submitted to Baal in Elijah's day.
Add to this the fact that Elisha asked for and received a double portion of the Spirit of Elijah. In 2 Kings 2:
9 When they had crossed, Elijah said to Elisha, “Tell me, what can I
do for you before I am taken from you?”
“Let me inherit a double portion of your spirit,” Elisha replied.
10 “You have asked a difficult thing,” Elijah said, “yet if you see me
when I am taken from you, it will be yours—otherwise, it will not.”
Elisha was Elijah's direct successor.
However, the strongest evidence comes from Jesus. As the Son of God, his attitude toward the written scriptures is the one we should have. In Matthew 5, Jesus says:
17 “Do not think that I have come to abolish the Law or the Prophets;
I have not come to abolish them but to fulfill them. 18 For truly I
tell you, until heaven and earth disappear, not the smallest letter,
not the least stroke of a pen, will by any means disappear from the
Law until everything is accomplished.
If you follow every place where Jesus quotes the Old Testament, or makes judgments based on its reasoning, you will find his endorsement everywhere. He never challenged one thing as false, though he did correct misinterpretations. He quoted from or referenced most books of the Old Testament, meaning that we can trust them, too.
See http://www.bible.ca/b-canon-old-testament-quoted-by-jesus-and-apostles.htm where it says that Jesus quoted from 24 of the Old Testament Books.