As a foreign correspondent, you learn not to believe what countries say about themselves. Don't base your reporting on headlines or campaign speeches meant for domestic consumption: They express not the country's genuine circumstances, only its national obsessions. France is always in crisis. Britain is forever in decline. And the United States, every four years, discovers that something terrible has happened to its once-great society, that it needs to become great again.

Americans and their institutions do face genuine challenges. Just not the ones you've been hearing in Cleveland.

You've been hearing that the United States has a big problem with illegal immigrants, big enough to necessitate building a wall, because the people coming from Mexico and Central America are inordinately prone to commit rape, murder and other violent crimes. "We have people pouring in – they're pouring in," Donald Trump repeats.

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In fact, undocumented immigration to the United States is at its lowest level since the 1960s. According to the Customs and Border Protection Agency, the number of migrants caught trying to cross the Mexico-U.S. border in 2015 was down by 18 per cent from the year before, at its lowest rate since 1969. And the number of undocumented people living in the United States has been falling continuously since 2008, from 12 million to 10 million. The reasons are largely economic: People go to the United States when there are job or business opportunities, and those have been weak of late.

Not that those undocumented U.S. residents are the danger they're made out to be: They commit crimes, and especially violent crimes, at a lower rate than average Americans do. A study this year found that the incarceration rate in the United States for foreign-born men under 40 was 1.6 per cent, less than half the rate (3.3 per cent) for native-born Americans. In other words, an increase in "illegal" immigration actually lowers the crime rate.

You've been hearing that "crime is out of control" in the United States. Speakers at the Republican convention repeatedly refer to "what is going on in our inner cities" – implying that crimes committed by black people and other minorities are especially rampant.

In fact, the United States is experiencing the lowest crime rates it has seen in a quarter century, and probably the lowest violent-crime rate in its history. In 2014 (the last year for which statistics are available), the violent-crime rate was 366 crimes for every 100,000 people – less than half the rate of America's most violent year, 1991, when it hit 758 for every 100,000, and now down to record-low levels of the pre-1970 years. The murder rate is down to 4.5, lower than most years in the 1950s and sixties.

And violent-crime rates in big cities such as Los Angeles, New York and Washington have dropped even more, to a third of their 1970s levels. Crime among black Americans has fallen even faster: Among African-American youth, according to the FBI, the past 20 years have seen rates of assault fall by 59 per cent, rape by 66 per cent and murder by 82 per cent.

The United States does face rising crime in one specific area: the mass-casualty shooting. Last year, there were 372 such atrocities (in which at least four people are shot), up from only a handful in the 1960s. This phenomenon (of which the terrorism "war" is a minor subset) is rooted in one policy problem not being addressed this week: the easy and largely uncontrolled availability of firearms.

And you've been hearing that the United States is in an economic crisis created by out-of-control taxation and government spending.

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In fact, central bankers are warning Washington that its spending has fallen to dangerous lows. Spending by all government levels on expenditures, infrastructure and investment (that is, excluding non-negotiable transfers such as pensions and Medicare) fell last year to their lowest point in 66 years after plummeting through the Obama years, leading Federal Reserve chair Janet Yellen to complain that fiscal policy needs to be more generous to get the economy back on track.

That lack of tax-and-spend clout is keeping other serious U.S. problems, such as the decade-long stagnation in middle-class incomes, from being addressed properly. But you sure didn't hear anything about that from Cleveland: There were plenty of bold solutions, but not to America's actual problems.