"argue through" is a correct phrase and can be used in written English.
It is used to mean debating or speaking about a topic in order to persuade someone of your opinion. For example: "We argued through our disagreement for a long time, but eventually we reached a conclusion."
We will always argue through our ideas but we know when to give up on something.
Effective systems leadership means building the trust and evidence to argue through to the right conclusion for the local health economy as a whole, not trying to reach a compromise.
But while reform-minded politicians typically focus on shortening sentences and improving living conditions, Stanley, Smith, and the dozens of others who contributed to "Captive Genders" argue, through historical analysis and personal experience, that the prison system should be abolished altogether.
We argue, through the case of the microprocessor industry, for greater attention to investment as an inter-firm and inter-agency process, thus lessening the fixation in studies of capital budgeting on the traditional hierarchical and bounded organization.
As the theater staff tried to remove the belligerent Palmer, he continued to argue through two more songs.
Businesses absorb higher labor costs, they argue, through reduced turnover, greater employee productivity and some price increases passed on to consumers.
Robin and I were arguing through the press.
Mr. Esser argued through most of the negotiations that Mannesmann would be better off alone.
It's a perception changer in other ways too, as it argues, through demonstration, against basic misunderstandings surrounding this art.
She was an explosive presence in the courtroom, crying and arguing through much of the nine-day trial.
He argues, through a campaign staff member, that the deal foundered because the Irish economy imploded in 2008.
