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The U.S. Capitol is seen during a rehearsal for the inauguration ceremony of U.S. President-elect Donald Trump.CARLOS BARRIA/Reuters

In the years before he became the American president in 1881, James Garfield employed two small right triangles inside a trapezoid to prove the Pythagorean Theorem in a fresh way. In the years to follow, many American presidents have had things to prove – Harry Truman, Lyndon Johnson and Gerald Ford to prove they were worthy to succeed presidents who by death or resignation left office early, Ronald Reagan and George W. Bush to prove they had the intellectual discipline for the presidency, Jimmy Carter and Bill Clinton to prove that their experience as governors of Southern states were sufficient for White House leadership.

Now comes Donald Trump with an unusual menu of things he must prove. The real estate and casino tycoon is not without experience or exposure – he's perhaps the best-known businessman in the country, perhaps in the world – but the 45th president will be the first chief executive to have no political, government or military leadership experience.

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And so in the days before he takes the oath of office at the West Front of the Capitol at noon on Jan. 20 – and in the weeks that follow, including a high-profile appearance before Congress, likely before his first month is completed – Mr. Trump surely is contemplating these presidential challenges:

He must prove to his supporters that he has their interests at heart. A billionaire who ran as a populist, Mr. Trump cultivated white, working-class voters in states such as Pennsylvania, Michigan and Wisconsin that pollsters and political professionals considered out of reach for him. He did so by speaking in an idiom of jobs and with great respect for police officers, factory workers and other blue-collar voters out of work and out of trust for government.

That was an audacious strategy, but the audacity of it speaks to the difficulty of redeeming those promises. He has made symbolic steps already, jawboning an Indiana company not to move air-conditioning workers' jobs to Mexico, hectoring General Motors Co. for importing cars from Mexico and criticizing the Ford Motor Co. for planing a $1.6-billion (U.S.) plant in Mexico that the auto maker backed away from early this year. These manoeuvres won a a wan critique; The Wall Street Journal among others, including Democrats, warned that discrete appeals or demands to individual companies could have deleterious results, perhaps discouraging firms from investing in the United States out of fear that Washington would intervene if it made a rational business decision later.

Mr. Trump promised jobs – not just jobs, but jobs for miners and steel workers – and now he has to deliver. Many business analysts believe this will be a Herculean task. A man who has for months talked big, now has to deliver in a big way.

He must prove to his detractors that he can be a credible president. Many of his rivals and critics not only doubt his credibility but also doubt his ability to manage a complicated government that is far bigger than any company he has ever run, with difficulties far more complex than any he has encountered.

This concern is buttressed by worries about the new president's confidence (too much), competence (too little) and comportment (too extreme). Many of these critics regard communication-by-tweet more the result of compulsion than of contemplation. They worry, moreover, that while he may have an affinity for Vladimir Putin or Russia, his charm will be ineffective if not repellent to other major leaders, including Justin Trudeau, who may regard him more as a batty uncle than a formidable thinker.

Still, Mr. Trump is an international figure already, is now (or soon will be) the leader of the world's most formidable superpower and is doubtless a global trendsetter. His caucus and primary victories presaged the Brexit vote in Britain, and now the prospects for populist, right-leaning political movements in Europe are brighter than they have been in decades.

He must prove to vital Clinton interest groups that he is not a danger to them. Resist this as he might, Mr. Trump cannot govern a diverse country by mocking diversity or dismissing it as a manifestation of political correctness. He said on Election Night that he planned to be president of all Americans, and that by definition includes African-Americans (whose home neighbourhoods he belittled), the disabled (one of whom he ridiculed in public), Hispanics (some of whom he has disparaged or threatened to deport) and Muslims (some of whom have been part of the American story since they arrived with slave ships in the 17th century).

He must prove to the country that his campaign vows aren't meaningless. This will be a difficult task. He has promised a wall on the country's Southern border, for example, but has given scant hints as to how that will be built, and when, and exactly where. Nor has he produced a plausible way to force Mexico to pay for the wall; that country's leader, Enrique Pena Nieto, tweeted after his fall meeting with Mr. Trump that he "made it clear that Mexico will not pay for the wall." But last week, Mr. Trump insisted that, one way or another, Mexico will pay. "That will happen, whether it's a tax or whether it's a payment – probably less likely that it's a payment – but that will happen," he said in his Wednesday press conference.

Mr. Garfield didn't prove the Pythagorean Theorem with tricks; he began his argument by asserting ½ab + ½ab + ½c2 = ½(a + b)(a + b), leading to a five-step proof that the Mathematical Association of America still describes as "a mathematical treasure." His successor 25 presidents later likewise must employ precision and avoid facile tricks as he struggles to prove himself as well.