'Sleeve' is PIE *sleubh- (to slip) — literally 'the tube the arm slips into.'
The part of a garment that wholly or partly covers a person's arm.
From Old English 'slīefe,' 'slȳfe' (sleeve), from Proto-Germanic *slaubō (a covering that slips on, a slip-on covering), from the verb *slaubaną (to slip, to slide into), related to *slūpaną (to slip, to glide). The PIE root is *sleubh- (to slip, to slide, to glide). A sleeve is literally 'a slip-on' — a tube into which the arm slides. The connection to smooth sliding explains the garment feature: a sleeve is the part you slip your arm through. The same PIE root *sleubh- produced 'slip,' 'slippery,' and 'slop' (wet and slippery). Germanic
'To have something up your sleeve' — to conceal a secret advantage — refers to the medieval practice of hiding objects in wide sleeves. And 'to wear your heart on your sleeve' comes from medieval jousting, where a knight would tie a lady's favor (ribbon or handkerchief) to his arm — literally displaying his romantic attachment on his sleeve for all to see.