It says one can know they possess eternal life.
Here is the verse:
1 John 5:13
New International Version
I write these things to you who believe in the name of the Son of God so that you may know that you have eternal life.
1. This verse does not even mention the Book of Life.
2. The verse is not written as a personal letter such as Philemon. Hence no one can claim that John was speaking directly to a certain individual or individuals.
3. The verse is speaking to Christians in general but not affirming which individuals are Christians.
4. Using your logic,
anyone can "assert" that their name is written in the Book of Life, by simply claiming you are part of the group that John is writing to.
Simply put, John is affirming that those who hold firm to the Christian faith will be saved but he is most certainly not guaranteeing an individual can know that they are holding firm to the Christian faith.
Remember, where false assurance exist true assurance cannot. Matthew 7:22