The connection of 1 Peter 5:12–14 to the preceding
1 Peter 5:12–14 (ESV)
12By Silvanus, a faithful brother as I regard him, I have written briefly to you, exhorting and declaring that this is the true grace of God. Stand firm in it.
As is evident from the doxology of 1 Peter 5:11, the apostle Peter has completed the argument (and hence purpose) of his letter. This closing paragraph contains the typical items one may expect as a writer rounds off his letter.
12 By Silvanus, a faithful brother as I regard him, I have written briefly to you, exhorting and declaring that this is the true grace of God. Stand firm in it.