"It is no surprise fewer and fewer Americans are achieving their dreams and more and more parents are concerned their children won't realize theirs," said Santorum, who challenged Mitt Romney to be the Republican presidential nominee.
Earlier Tuesday, Romney and Rep. Paul Ryan of Wisconsin were nominated as the party's presidential and vice presidential candidates.
"Under President Obama, the dream of freedom and opportunity has become a nightmare of dependency with almost half of America receiving some government benefit," Santorum said during the evening session of the convention's first day.
He spoke of his father, who emigrated from Italy and "didn't come here for some government guarantee of income equality or government benefits to take care of his family."
During his presidency, Obama has borrowed $5 trillion while "trying to convince you that he could make things better for you -- to put your trust in him and the government to take care of every problem," Santorum said.
"The result -- massive debt, anemic growth and millions more unemployed," he said. "The president's plan didn't work for America, because that's not how America works."
If America is going to succeed, "we must stop the assault on marriage and the family," he said.
"The fact is that marriage is disappearing in places where government dependency is highest," Santorum said.
Americans believe in freedom and the responsibility that comes with it to work hard to ensure that everyone can reach their potential, he said.
"We believe it because it still works," Santorum said.
"A vote for Mitt Romney and Paul Ryan will put our country back in the hands of leaders who understand what America can," he said, "and, for the sake of our children, must be to keep the dream alive."
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