1John 5:13
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.
does eternal mean temporary?
You are correct eternal does not mean temporary. However, the words “:you have eternal life” come after the words “may know”. These words (may know) does not tell the reader they have absolute certainty of eternal life. These words here are used the same way we use them today when we say I know I am going to win first place in tomorrows race. There is no way we can have absolute certainty that we won’t twist our ankle half way through the race and lose. Just like in the verse here we are using the words “may Know” to mean we have confident assurance in the outcome, not infallible certainty. St. John uses the same words “may know” 2 verses later when he says “we know that we have obtained the requests made of him”. If St. John is telling us we have absolute certainty here that God will answer our prayer, why would he then go and tell us, but don’t pray for mortal sin because God won’t answer those prayers. Wait didn’t he just say “we know” we will obtain any request asked?
2 Peter 2:10
Therefore, my brothers and sisters, make every effort to confirm your calling and election
Is it possible the elect will not be saved at the end?
I think you mean 2 Peter 1:10, typos happen no biggie.
If we back up to the beginning of the chapter we see St. Peter must be speaking to the elect when he says “who have obtained a faith of equal standing with ours” in verse 1. Now when we read through to his climactic statement in verse 10 we see the thing the elect do. They:
supplement your faith with virtue
and virtue with knowledge
and knowledge with self-control
and self-control with steadfastness
and steadfastness with godliness
and godliness with brotherly affection
and brotherly affection with love.
St. Peter goes onto tell us in verse 9 that “whoever lacks these things is blind and shortsighted and has forgotten that he was cleansed from his old sins.”
In verse 9 he is telling us that we are the elect (cleansed from our old sins) however he let’s it be known that it is possible to forget this election if we lack these things. The only way to forget it is if you already had it.
It is in this context that St. Peter says “Therefore, my brothers and sisters, make every effort to confirm your calling and election”.
Confirming our election is the only way we can have confidence that we might be saved. St. Peter even tells us in the next line that we can not have assurance of being saved when he says “for if you do this you will never fall”. Which would mean that if we don’t do these things there is a possibility that we, the elect" might fall. And we will not recieve what is promised in verse 11. “so there will be richly provided for you an entrance into the eternal kingdom of our Lord and Savior Jesus Christ.”