Standard Christian belief is that we are saved via the holiness of Christ ...
2 Corinthians 5:21 He made Him who knew no sin to be sin on our behalf, so that we might become the righteousness of God in Him.
So that we MIGHT become the righteousness of God in Him. The righteous way in which we live can apply Christ's sacrifice (See: 1 John 1:7). Also, keep reading in 2 Corinthians and you will see that Paul talks about how they sought forgiveness with the Lord over their recent sin (2 Corinthians 7:10), and yet Paul feared that they would have not repented of their sins.
"Again, think ye that we excuse ourselves unto you? we speak before God in Christ: but
we do all things, dearly beloved, for your edifying. For I fear, lest, when I come, I shall not find you such as I would, and
that I shall be found unto you such as ye would not: lest
there be debates, envyings, wraths, strifes, backbitings, whisperings, swellings, tumults:
And lest, when I come again, my God will humble me among you, and
that I shall bewail many which have sinned already, and have not repented of the uncleanness and fornication and lasciviousness which they have committed." (2 Corinthians 12:19-21).
Paul says for them to examine themselves on whether they are in the faith unless they are reprobate (2 Corinthians 13:5).
Why would they need to examine themselves if they are simply saved by having a belief alone on Jesus?