This is talking about the general atonement made for all of mankind.
Unless you believe in Universal Salvationism, the atonement is applied when we have true faith. True faith starts off in believing and or receiving Jesus as one's Savior (whereby the atonement is applied). But after coming to Jesus by faith, we have to continue in His Word, continue in his love, continue in the faith of obeying Jesus in order for the atonement to continue to be applied to our life. Works of faith are just evidence that a person's faith is true or genuine. James says he will show you his faith by his works (James 2:18). The author of Hebrews says that without holiness, no man shall see the Lord (Hebrews 12:14). Paul says that a person can deny God by a lack of works (Titus 1:16). So one has to have the right kind of faith in order for the atonement to be applied.
How can we prove it?
Hebrews 5:9 says,
"And being made perfect, he became the author of eternal salvation unto all them that obey him;"
1 John 1:7 says,
"But if we walk in the light, as he is in the light, we have fellowship one with another, and the blood of Jesus Christ his Son cleanses us from all sin."
Poor word choice on my part. I agree. The blood of Christ takes away sin. But there is a cleansing going at some point for 1 John 1:7 says if we walk in the light as He is in the light, the blood of Jesus cleanses us from all sin. Meaning, if we love our brother (we abide in the light - See 1 John 2:9-10) (Note: Paul says to love our neighbor is the equivalent of keeping the Moral Law like: Do not murder, do not steal, and do not covet, etc. - See Romans 13:8-10).
In Hebrews 10:4, we learn that the blood of goats and bulls never took away sin. They only temporarily atoned or covered sin because they had to keep revisiting the same PAST sins year after year. But when Jesus came and died for us (giving us the perfect sacrifice) He is able to provide a way that is more perfect in the fact that all our PAST sins are forgiven for good. They are remembered no more.
But if you were to keep reading, this does not include willfully sinning again after receiving the knowledge of the truth.
For Hebrews 10:26 says,
"For if we sin wilfully after that we have received the knowledge of the truth, there remaineth no more sacrifice for sins,"
So again, a person needs a true faith in order to access the saving grace of God and willfully sinning after receiving the truth is not true faith because there remains no more sacrifice for sin if we willfully sin again.
This is merely a matter of deflection away from the problem of Eternal Security.
The problem is that when an Eternal Security Proponent says that King David was saved while he committed his sins of murder and adultery, they are telling other people that they can be like a David and do these kinds of sins and be saved, too. For if David did it, then they surely can do that, too.
You missed the second half of the verse. It says that you may receive MERCY and FIND GRACE to help us in our time of need. IF mercy and grace was already in our possession there would be no need to approach God's throne and seek out our Heavenly high priest who is Jesus Christ.
"And if any man sin, we have an advocate with the Father, Jesus Christ the righteous:" (1 John 2:1).
"If we confess our sins, he is faithful and just to forgive us
our sins, and to cleanse us from all unrighteousness." (1 John 1:9)
This is in view that a person is to "sin not" (1 John 2:1) and to walk in the light so as to be cleansed of all sin by the blood of Jesus (1 John 1:7).
For he that confesses and forsakes sin shall have mercy (Proverbs 28:13).
Jesus said that the Ninevites will rise up in judgment against this generation because they repented at the preaching of Jonah (Matthew 12:41). If you were to turn to Jonah 3:6-10, you would learn that the King of the Ninevites told his people to do two very important things.
(a) Cry out to God (repentance).
(b) Forsake their evil ways (i.e. the fruits of repentance).
Now, when God had seen that they turned from their wicked ways, THAT was the point where God decided not to bring wrath and destroy them anymore and not before. So yeah. The whole "trust in the finished work of Christ alone" for salvation is not something even remotely taught in the Bible. James says we are justified by works and not by faith alone (James 2:24). For works of faith are merely the proof in the pudding that one's faith in Christ is true.