arm-band n.

[upper limb of the human body], Middle English arm, from Old English earm, from Proto-Germanic *armaz (source also of Old Saxon, Danish, Swedish, Middle Dutch, German arm, Old Norse armr, Old Frisian erm), from PIE root *ar- "to fit together" (source also of Sanskrit irmah "arm," Greek arthron "a joint," Latin armus "shoulder").

Arm of the sea was in Old English. Arm-twister "powerful persuader" is from 1915. Arm-wrestling is from 1899.

They wenten arme in arme yfere Into the gardyn [Chaucer]

"a flat strip," also "something that binds," Middle English bende, from Old English bend "bond, fetter, shackle, chain, that by which someone or something is bound; ribbon, ornament, chaplet, crown," with later senses and spelling from cognate Old Norse band and technical senses from Old French bande "strip, edge, side" (12c., Old North French bende), all three ultimately from Proto-Germanic *bindan, from PIE root *bhendh- "to bind."

The meaning "a flat strip" (late 14c.) is from French. In Middle English, this was sometimes distinguished by the spelling bande, bonde, but with loss of terminal -e the words have fully merged via the notion of "flat strip of flexible material used to wind around something."

The meaning "broad stripe of color, ray of colored light" is from late 14c.; the electronics sense of "range of frequencies or wavelengths" is from 1922. Most of the figurative senses ("legal or moral commitment; captivity, imprisonment," etc.) have passed into bond (n.), which originally was a phonetic variant of this band. The Middle English form of the word is retained in heraldic bend (n.2) "broad diagonal stripe on a coat-of-arms."