It's one thing to become converted, to go through a conversion experience. It's entirely another to stay that way. Conversion is not a one-time event. In order to stay converted, you need to continue to live your faith and work at it.
After King Lamoni's father was converted, he enacted some laws that really allowed for religious freedom in his kingdom. He made sure that laws were in place that made it so people could preach and teach and practice their religion. Ammon and his brothers could now go throughout the land teaching and sharing the Gospel with the Lamanites, with great success. The scriptures teach that thousands were converted.
So great and deep was their conversion that this is what is said about them:
"And as sure as the Lord liveth, so sure as many as believed, or as many as were brought to the knowledge of the truth, through the preaching of Ammon and his brethren, according to the spirit of revelation and of prophecy, and the power of God working miracles in them—yea, I say unto you, as the Lord liveth, as many of the Lamanites as believed in their preaching, and were converted unto the Lord, never did fall away." (Alma 23:6)
For it to be said that they "never did fall away", they must have changed so much. They put away their weapons of war, and no longer fought. Lamanites, historically, were a more wicked and bloodthirsty people. Not all Lamanites were converted, but so great was this peoples' conversion that they even wanted to change their name to be able to be known by something else. They decided from then on to be called the Anti-Nephi-Lehies. They worked hard, and lived the gospel, and they were a good people.
I feel like it would be great if, at the end of my life, it could be said of me "he never did fall away". I still have a long way to go, but I do desire to have that depth of conversion. I want to do what is right, and help people along the way.