Philippians 1:6

‘Being confident of this very thing, that he who began a good work in you will perfect it until the day of Jesus Christ,'

And his prayers were especially aided by the confidence that he had that the God Who had begun a good work in the Philippians, and Who was working in them to will and to do of His good pleasure (Philippians 2:13), would make that good work perfect and complete until the Day of Jesus Christ. For while he could exhort God's people to faithfulness as urgently as anyone, Paul never had any doubt that, in those who were truly His, God would complete His perfect work. Compare for this 1 Corinthians 1:8; 1 Peter 1:5; Jude 1:24; John 10:27. And he knew that this must be so, for we who are His are God's personal gift to Jesus Christ, over whom He constantly watches and prays (John 6:37; John 6:39; John 10:29; John 17:6; John 17:9; John 17:24; Hebrews 7:25). Jesus Christ is our perfect and complete Saviour from start to finish (Hebrews 2:10), and not one of His own will be lost.

For ‘doing a good work' in someone we should compare Matthew 26:10. It there refers to the bringing about of an inward personal experience that is spiritually beneficial. The same idea is in mind here. And this work would continue, until it was finally brought to completion in the Day of Jesus Christ. The ‘day of Jesus Christ' is that day when He will call his own to give account and will, having perfected them, bring them into His eternal kingdom (see Philippians 1:10; Php 2:16; 1 Corinthians 1:8; 1 Corinthians 5:5; 2 Corinthians 1:14).

The same combination of ‘begin' and ‘perfect, make complete' is found in Galatians 3:3 where the work begun by the Spirit is seen as requiring to be completed by the Spirit (as Paul says, it is absurd to think that it will be completed by the flesh). Thus we have in both contexts the idea of God actively beginning and completing His work, in Galatians 3:3 by His Spirit (compare Philippians 1:27).