The proof (την δοκιμην). "The test" as of metals (2. Corinthians 2:9; 2. Corinthians 9:13). Three times they had seen Timothy (Acts 16:13; Acts 19:22; Acts 20:3).

With me (συν εμο). Paul's delicacy of feeling made him use συν rather than εμο alone. Timothy did not serve Paul.

In furtherance of (εις). See Philippians 1:5 for this use of εις.