Determined individual finally loses heart (7)
I believe the answer is:
sticker
'determined individual' is the definition.
(someone who sticks at something)
'finally loses heart' is the wordplay.
'finally' suggests the final letters.
'heart' becomes 'ticker' (synonyms).
The final letter of 'loses' is 's'.
's'+'ticker'='STICKER'
(Other definitions for sticker that I've seen before include "Gummed label" , "Person with staying power" , "problem" , "Adhesive label" , "he soldiers on!" .)