Copenhagen — The world is coming to know President Barack Obama, the pragmatist whose stand at a messy global warming summit underscored the way he leads: Let’s get done what we can, imperfect as it is.

When Obama impatiently told other leaders here to rally behind a climate change deal despite its limitations, he sounded like he was talking about his health care push or his economic stimulus plan at home. The president whose election campaign was about change knows that governing — and re-election — are about showing results.

“I’m sure that many consider this an imperfect framework,” Obama said Friday to a gathering of leaders from 193 countries. “No country will get everything that it wants.”

Mark Garten/AP Photo

U.S. President Barack Obama leaves the podium after addressing the High Level Plenary meeting on Climate Change in Copenhagen in this picture provided by the United Nations.

On several levels, watching Obama in Copenhagen was like getting a mini-course in what makes him tick as a leader.

He’s learning the frustrating limits of his powers of persuasion. Yet he presses on, willing to jet across the ocean and plunge into a long day of unpredictable diplomacy in pursuit of a deal to fight global warming.

At the same time, he’s hemmed in by the high expectations he helped create. Across the world, so many people have come to see Obama the way they want that his actions often don’t fit their perceptions, and he finds it ever harder to meet people’s wishes.

He won the White House because voters from all over the U.S. political map placed in Obama their hopes for the future — expanding health care all around, reducing racial tensions, saving the planet. Now his approval ratings hover around 50 percent, down considerably from the heady, early days. Reality is harder than hope.

And so when Obama plunged at the last minute into a summit where the best chance was a political deal — talk of a legally binding pact had long been quashed — he sought to grab the moment and claim what he could.

“We can embrace this accord, take a substantial step forward, continue to refine and build upon its foundation,” Obama said. In other words, focus on what’s in, not out, and don’t walk away empty-handed.

Later, after a long day of negotiations, Obama announced that the U.S., China and three other countries had reached an “unprecedented breakthrough” on curbing climate change. Yet not without more hard work and trust would a more comprehensive, binding deal be in sight, he added.

This was only a start, but one that was well worthwhile.

“This is a classic example of a situation where if we just waited for that (legal deal), then we would not make any progress,” he said. “There might be such frustration and cynicism that rather than taking one step forward, we ended up taking two steps back.”

So it goes across the many fronts of Obama’s agenda. He is no longer on his own pace to sign a nuclear arms deal with Russia, or to close the Guantanamo Bay prison, or to get a health insurance overhaul through Congress. But there is halting movement on them all, and that’s what he likes to talk about.

In Copenhagen, Obama’s diplomacy displayed the importance of his presence and the limits of his power. He has promised a U.S. posture of leading without lecturing, although he did both here as his patience shrank.

Obama again showed his style of making a direct appeal at a big event, which carries risk and reward.

Though negotiators worked through the night, Obama and other leaders arrived here Friday morning for the U.N. climate conference with hopes for a deal dimming instead of brightening.

Many officials in Copenhagen have been laboring for years over extraordinarily complex issues. But Obama, not even a year into his job at the helm of the world’s most powerful nation and its second-largest polluter, seemed to say it was time to move on.

“After months of talk, and two weeks of negotiations, after innumerable side meetings, bilateral meetings, endless hours of discussion among negotiators, I believe that the pieces of that accord should now be clear,” he said, one of several impromptu flashes of impatience he injected into his brief remarks.

Yet the negotiations didn’t end just because Obama said they should. He said America was showing responsibility and taking bold action on climate, but critics sniffed at that.

Obama’s approach is to grab wins where he can, emphasizing progress even when goals are scaled back and deadlines missed. And so he went to Copenhagen to try to close a deal. It was about results.

Even mixed and modest ones.

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