We aren't inherently immortal no. God kicked Adam and Eve out of the garden because He didn't want them to eat of the tree that would give them eternal life. Ergo, it wasn't there before.
Discussion
That sounds crazy to me that you don't believe in life everlasting. My first thought is if God loves us so much why doesn't he want us around with him forever?
Those who eat of the tree and who's names are in the Lamb's book of life (people who have a relationship with Him through Jesus and the Holy Spirit and faith) have eternal life with Him for all of eternity. Sorry, apparently I wasn't clear. I meant I don't believe all people are inherently eternal beings, it must be given by God. All else perish.