If someone does wrong to us but remains unrepentant and won't admit their sin, how do we show them forgiveness, without encouraging them to continue sinning in the same way against us?
The Biblical expectation for forgiveness is that we forgive those who repent and seek our forgiveness:
Luke 17:3 “ Be on your guard! If your brother sins, rebuke him; and if he repents, forgive him.
Luke 17:4 “And if he sins against you seven times a day, and returns to you seven times, saying, ‘I repent,’ forgive him."
The Father in Heaven insists that we forgive all who seek our forgiveness, otherwise He will not show mercy to us in our sins:
Matt. 6:14 “ For if you forgive others for their transgressions, your heavenly Father will also forgive you.
Matt. 6:15 “But if you do not forgive others, then your Father will not forgive your transgressions.
Jesus says that if we are not inclined to forgive others when they ask us, then our Father in Heaven will not forgive us our transgressions. Jesus is not referring to the eternal forgiveness that comes to all believers at the point of faith in Christ. This forgiveness is not conditional. All believers in Jesus receive forgiveness of sins and will not suffer the Second Death reserved for unbelievers.
Instead, Jesus is speaking about the Father's willingness to forgoing bringing earthly consequences upon us for the sins we commit in our life as believers. If we are so hard-hearted that we never forgive others who request our forgiveness, then when we sin on earth, the Father in Heaven will bring the consequences of our sin upon our heads to discipline us and make a point. Our incentive to be forgiving to those who ask is to remember that the Father forgave us when we asked.
What do we do for the believer who doesn't ask for forgiveness? The Biblical definition of forgiveness is a grant of mercy to the one who seeks it, so the process of forgiveness begins with the person who has committed the transgression asking for forgiveness. If that request is never made, then the offended person has no way to communicate forgiveness. You may choose to love the trangressor in spite of his sin and hold no grudge against him in your heart (which is a good and proper thing to do), but technically speaking, you cannot forgive him until he opens the door for the opportunity.
If a believer is unaware of his own sin or unwilling to admit his sin or blames others for his mistakes, then he is not seeking forgiveness and you are not able to offer it. You can't extend forgiveness to a person who doesn't seek it, but this doesn't mean you should hold a grudge or treat him with contempt. Instead, show him grace and love him despite his wrong thinking, and gently work to show him his error, preferably by teaching him from the Bible.
Some personalities are prone to defensiveness and a poor ability to see themselves honestly, which makes working with them difficult under these circumstances. Nevertheless, the Lord is your hope to open their eyes. When this happens, they will ask for your forgiveness, and you shall grant it.