The inference is that the detail that best supports the speaker’s purpose to convince the audience of Athens’s power and influence is "everywhere we have left behind us everlasting memorials”.
It should be noted that the inference simply means the conclusion that can be deduced based on the information that given in the story.
From the information, it was stated that future ages will wonder at us, as the present age wonders at us now. We do not need the praises of a Homer, or of anyone else whose words may delight us for the moment, but whose estimation of facts will fall short of what is really true.
Therefore, the inference is that the detail that best supports the speaker’s purpose to convince the audience of Athens’s power and influence is "everywhere we have left behind us everlasting memorials”.
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