<p>In 1508, the Spanish explorer Juan Ponce de Leon----the same Ponce de Leon who later would seek the fountain of youth----landed on Puerto Rico( accompanied by a small boat).
why would seek has no error, I am confused.</p>
<p>nothing wrong with “would seek”. Would indicates future action that happened in the past.
First, Juan Ponce de Leon landed on Puerto Rico
Then, in the future [see LATER–> indicate future action] he seek the fountain of youth.
Therefore, it is correct to use would+seek; it helps the reader to see clearer relationship between two actions that happened in the past.</p>