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Show off, broadcast succeeded (3,2,4)

Ross

I believe the answer is:

put on airs

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'show' is the definition.
Although both the answer and definition are verbs in their base form, I can't understand how they can define each other.

'off broadcast succeeded' is the wordplay.
'off' becomes 'puton' (I can't explain this - if you can you should give a lot more credence to this answer).
'broadcast' becomes 'air' (to air on TV).
'succeeded' becomes 's' (genealogical abbreviation).
'puton'+'air'+'s'='PUT ON AIRS'

Can you help me to learn more?

(Other definitions for put on airs that I've seen before include "assume a haughty manner" , "DJ might" , "A deejay might" , "act pretentiously" , "Behave in affected and haughty manner" .)

I've seen this clue in The Times.
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