Definition of fain adverb from the Oxford Advanced Learner's Dictionary

 

fain

 adverb
adverb
BrE BrE//feɪn//
 
; NAmE NAmE//feɪn//
 
(old use)
 
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willingly or with pleasure I would fain do as you ask. Word Origin Old English fægen ‘happy, well pleased’, of Germanic origin, from a base meaning ‘rejoice’; related to fawn ‘to try to please someone by paying them too much attention’.
See the Oxford Advanced American Dictionary entry: fain