"So it would have served no purpose."
Translation:Alors cela n'aurait servi à rien.
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My long answer has vanished!
"servir à rien" is a phrase that means "be of no use"
To quote from another post:
the verb servir can be both transitive and intransitive. As a transitive verb it can mean 'to serve' and as an intransitive verb it can mean 'to be useful'.
If you use servir as an intransitive verb then
servir à quelque chose - to be useful for something.
servir de quelque chose - to be used as something.
I also do not think that you sentence is grammatically correct French. After looking for examples I found it was continuously changed to the answer above. To see what I mean go to https://www.linguee.com/french-English
and type in n'aurait rien servi to see what I mean.