Here in Chamonix, the air finishes much colder (5)
I believe the answer is:
icier
'much colder' is the definition.
'here in chamonix the air finishes' is the wordplay.
'here in chamonix' becomes 'ici' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'finishes' suggests the final letters (I've seen 'finish' mean this).
The last letters of 'the air' are 'er'.
'ici'+'er'='ICIER'
Can you help me to learn more?
(Other definitions for icier that I've seen before include "More cold - really freezing" , "much less friendly" , "rather cold" , "More glacial" , "More slippery" .)