So, how the heck can one Christian convince a nonbeliever? It isn't by using the Bible. That's getting the cart before the horse.
99% of the time, when witnessing, we're creating a relationship with another human being. We genuinely care about others, for Jesus said to do very specific things if we are His Sheep.
If an atheist, you could stop listening if the Bible offends you, for I'm preaching to the choir (the Christians) for a moment:
31“But when the Son of Man comes in His glory, and all the angels with Him, then He will sit on His glorious throne.32“All the nations will be gathered before Him; and He will separate them from one another, as the shepherd separates the sheep from the goats;33and He will put the sheep on His right, and the goats on the left.
34“Then the King will say to those on His right, ‘Come, you who are blessed of My Father, inherit the kingdom prepared for you from the foundation of the world.35‘For I was hungry, and you gave Me something to eat; I was thirsty, and you gave Me something to drink; I was a stranger, and you invited Me in;36naked, and you clothed Me; I was sick, and you visited Me; I was in prison, and you came to Me.’37“Then the righteous will answer Him, ‘Lord, when did we see You hungry, and feed You, or thirsty, and give You something to drink?38‘And when did we see You a stranger, and invite You in, or naked, and clothe You?39‘When did we see You sick, or in prison, and come to You?’40“The King will answer and say to them, ‘Truly I say to you, to the extent that you did it to one of these brothers of Mine, even the least of them, you did it to Me.’
41“Then He will also say to those on His left, ‘Depart from Me, accursed ones, into the eternal fire which has been prepared for the devil and his angels;42for I was hungry, and you gave Me nothing to eat; I was thirsty, and you gave Me nothing to drink;43I was a stranger, and you did not invite Me in; naked, and you did not clothe Me; sick, and in prison, and you did not visit Me.’44“Then they themselves also will answer, ‘Lord, when did we see You hungry, or thirsty, or a stranger, or naked, or sick, or in prison, and did not take care of You?’45“Then He will answer them, ‘Truly I say to you, to the extent that you did not do it to one of the least of these, you did not do it to Me.’46“These will go away into eternal punishment, but the righteous into eternal life.” Matthew 25: 34-46
When we do these things in a sincere selfless manner, as many missionaries have done in countless nations, then many other people seeing their good works will say, "Sir or Madam, why are you here doing all of these wonderfully kind things? It's expensive for you to be building schools to educate our children. That hospital is saving so many lives. Those wells you dug for us, we are getting healthier now that you have built them. Why do you do these things for us?".
Now the missionaries can say, "We do it out of love. We don't do it out of reason. We do it because once someone helped us and they were Christians. Jesus told us to do these things. He is our Master. He said it, so we do it. He must increase and we must decrease. We want to share that goodness from our Master with you, if you are interested. We will do these things even if you're not interested, for there is Free Will. It's up to you to listen if interested."
Many will listen then because we're doing what Jesus said to do. Jesus Saves...not us. We are frail beings and make mistakes. Jesus is undeniably kind hearted and good. Seldom will people say the opposite of Jesus.