"I've Tried to Believe But Can't. What Can I Do?" - John Piper's Answer Examined

I recently stumbled across John Piper's attempt to handle the following statement and question:

I've concluded that I don't have saving faith. I've tried to believe in Jesus for two years but I can't. I fear I'm beyond saving. What can I do?  

In Piper's attempt to answer this query he makes a number of theological assertions contrary to the notion of salvation by sovereign grace, but which are no-doubt frequently affirmed in Christendom today. What follows is a line-by-line, biblical analysis of Piper's answer:

Exercises in Right Division (Ezekiel 3:20)

Statement One:  Ezekiel 3:20 says, "Again, When a righteous man doth turn from his righteousness, and commit iniquity, and I lay a stumbling-block before him, he shall die: because thou hast not given him warning, he shall die in his sin, and his righteousness which he hath done shall not be remembered."  This verse is commonly interpreted to mean that an eternally saved man to lose his salvation through sinning.

Statement Two: The Lord Jesus Christ taught, "My sheep hear my voice, and I know them, and they follow me: And I give unto them eternal life; and they shall never perish, neither shall any man pluck them out of my hand." (John 10:27-28) Here the Lord is teaching that he gives a gift to a people he refers to as his "sheep" and that this gift is something called "eternal life" which involves a promise that they "shall never perish."  This statement is often used to support the notion that it is impossible for someone who once possessed eternal salvation to ever fall from grace.