Sorry, I checked, it really says these:
These will go away into eternal punishment, but the righteous into eternal life.
Mat. 25:46
For I tell you that unless your righteousness exceeds that of the scribes and Pharisees, there is no way you will enter into the Kingdom of Heaven.
Mat. 5:20
And righteous is know by actions:
He who does righteousness is righteous, even as he is righteous. He who sins is of the devil, for the devil has been sinning from the beginning. To this end the Son of God was revealed, that he might destroy the works of the devil. Whoever is born of God doesn't commit sin, because his seed remains in him; and he can't sin, because he is born of God. In this the children of God are revealed, and the children of the devil. Whoever doesn't do righteousness is not of God, neither is he who doesn't love his brother.
1 John 3:7-10
I am not making these up, they are really in the Bible, and I think people should respect what the Bible tells.
And about believing:
If anyone listens to my sayings, and doesn't believe, I don't judge him. For I came not to judge the world, but to save the world. He who rejects me, and doesn't receive my sayings, has one who judges him. The word that I spoke, the same will judge him in the last day.
John 12:47-48
Well, you didn't "check" very far, it appears.
Please check these:
John 3:15-16 15 so that whoever believes will in Him have eternal life. 16 "For God so loved the world, that He gave His only begotten Son, that whoever believes in Him shall not perish, but have eternal life.
John 3:36 "He who believes in the Son has eternal life; but he who does not obey the Son will not see life, but the wrath of God abides on him."
John 5:24 "Truly, truly, I say to you, he who hears My word, and believes Him who sent Me, has eternal life, and does not come into judgment, but has passed out of death into life.
John 6:40 "For this is the will of My Father, that everyone who beholds the Son and believes in Him will have eternal life, and I Myself will raise him up on the last day."
John 6:47 "Truly, truly, I say to you, he who believes has eternal life.
Rom 6:23 For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord
1 Tim 1:16 Yet for this reason I found mercy, so that in me as the foremost, Jesus Christ might demonstrate His perfect patience as an example for those who would believe in Him for eternal life.
Gal 3:22 But the Scripture has shut up everyone under sin, so that the promise by faith in Jesus Christ might be given to those who believe.
1 John 5:13 These things I have written to you who believe in the name of the Son of God, so that you may know that you have eternal life.
Summary: when one believes, they receive eternal life.
btw, righteousness is credited to those who believe. We don't earn it.