Note the words "continue to believe". That implies people who already believe. So is John trying to win converts or is he trying to provide a focus for the community to keep them together and united in faith in Jesus?
I think to understand this you need to see the saying in context ------
John 20:30-31
Amplified Bible
Purpose of Writing this Gospel
30
There are also many other signs (attesting miracles) that Jesus performed in the presence of the disciples, which are not written in this book;
31
but these have been written so that you may believe [with a deep, abiding trust] that Jesus is the Christ (the Messiah, the Anointed), the Son of God; and that by believing [and trusting in and relying on Him] you may have life in His name.
My view is
This is for Believers ---as there are Babes in Christ who are not mature in their Faith walk with Christ and John is speaking to encourage their Faith to believe and trust in the Words that are written in the Book -------as John here gives the purpose for the Gospel being written ------to encourage believing people who may have wavering Faith to continue to believe and trust in who Jesus is and rely on Him so
by their believing and trusting in the Word they have eternal life in His Name -----
that Jesus is the Christ (the Messiah, the Anointed), the Son of God; and that by believing [and trusting in and relying on Him] you may have life in His name.
Greek word for -----
continue--- which is the word you use in your Post
Strong's Exhaustive Concordance
menó: to stay, abide, remain
abide, continue, dwell, remain
A primary verb; to stay (in a given place, state, relation or expectancy) -- abide, continue, dwell, endure, be present, remain
to remain beside, continue always near
Greek word for--- May ---as used in Amp Bible
Strong's Concordance
echó: to have, hold
Definition: to have, hold
Usage:
I have, hold, possess.
So believers are to trust and believe that what the Word---Who Is Jesus ---- says is true and will come to pass -------God's Faith Produces what it says it will in His Word -------
here is what John is saying ---behind the words in my view ------
Don't allow the cares of this World or others who try to use their influence to distract your Faith in what is Written in this Book -----Believers have to remain Faithful in their belief and trust God's word and keep a close relationship with Jesus --and they will possess eternal life ------
This is a perfect example of God's Faith which is relying on what you can't see yet --- that is having eternal life in Heaven with Jesus -----all you have is the written Word and God's Faith to sustain your belief -------your Hope is in the future -----
