You may have heard the argument that since nowhere in the Bible is it directly stated that Lucifer is Satan, maybe he’s not.
However reading scripture with attention to the Holy Spirit's illumination of it as you do, you will see that it says so indirectly. Begin in ISAIAH 14:12,
"How you are fallen from heaven, O Lucifer, son of the morning! How you are cast down to the ground you who weakened the nations."
Now compare that declaration with later clarification in THE REVELATION OF JESUS CHRIST, Chapter 12 Verse 9,
"So the great dragon was cast out, that serpent of old, the Devil and Satan, who deceives the whole world; he was cast to the earth..."
Finally, thank the Spirit for proving to you that Lucifer is indeed that old devil, Satan!