Yes it is very true. The Bible tells us that we are new creatures in Christ. We are clothed in His righteousness...though we are not righteous. We have been sealed with the Holy Spirit. We have been adopted into the family. There was nothing I could do to merit salvation...there is nothing I can do to forfeit it. I can become rebellious...but, the Holy Spirit will wear me out if I am His.
Much of the mistake that occurs, is not that we teach eternal security for the believer. The problem is that many unbelievers believe they have eternal security based upon an emotional response to an altar call at a youth camp, conference, watching a Billy Graham crusade. They may have made a confession of faith, but if the tree never bore the fruit of the Spirit there is a good chance they weren't truly born again. Consequently, they have no motivation to give God the time of day.
But, certainly God doesn't un-adopt us. He doesn't un-make us a new creature in Christ. Every soul that has ever accepted Christ/became born again is in heaven or still walking this earth. The One who said he will never leave us or forsake us meant what he said. The One who said He is married to the backslider meant what He said. HALLELUJAH!!! He "did for me that that I could not do for myself". And, as BR Lakin would say, "There hangs a rainbow of hope on the shimmering shoulders of the dying storm of my bereavement". Because He lives...I can face tomorrow.