North Korea said on Saturday that firing its rockets at the U.S. mainland was "inevitable" after U.S. President Donald Trump called Pyongyang's leader "rocket man," in a further escalation of rhetoric between the two leaders.

North Korean Foreign Minister Ri Yong Ho's remarks before the United Nations General Assembly came hours after U.S. Air Force B-1B Lancer bombers escorted by fighter jets flew in international airspace over waters east of North Korea, in a show of force the Pentagon said demonstrated the range of military options available to Trump.

"Through such a prolonged and arduous struggle, now we are finally only a few steps away from the final gate of completion of the state nuclear force," Ri told the annual gathering of world leaders.

"It is only a forlorn hope to consider any chance that the DPRK (Democratic People's Republic of Korea) would be shaken an inch or change its stance due to the harsher sanctions by the hostile forces," he said.

Trump announced new U.S. sanctions on Thursday that he said allow targeting of companies and institutions that finance and facilitate trade with North Korea. Earlier this month the UN Security Council unanimously adopted its ninth round of sanctions on Pyongyang to counter its nuclear and ballistic missiles programs.

Ri, who said Pyongyang's ultimate goal was to establish a "balance of power with the U.S.," retorted that Trump himself was on a "suicide mission" after the U.S. president said Kim Jong-un was on such a mission.

Late Saturday night, Trump continued the war of words saying on Twitter that Kim and Ri "won't be around much longer" if Ri echoed the thoughts of "Little Rocket Man," an apparent reference to Kim.

Just heard Foreign Minister of North Korea speak at U.N. If he echoes thoughts of Little Rocket Man, they won't be around much longer! —@realDonaldTrump

Demonstration of U.S. resolve

The U.S. bombers' flight was the farthest north of the demilitarized zone separating North and South Korea that any U.S. fighter jet or bomber has flown in the 21st century, the Pentagon said.

North Korea's Foreign Minister Ri Yong Ho speaks during the 72nd session of the United Nations General Assembly on Saturday at UN headquarters in New York. (Julie Jacobson/Associated Press)

"This mission is a demonstration of U.S. resolve and a clear message that the President has many military options to defeat any threat," said Pentagon spokesperson Dana White, calling North Korea's weapons program "a grave threat."

"We are prepared to use the full range of military capabilities to defend the U.S. homeland and our allies."

Ri warned Pyongyang was ready to defend itself if the U.S. showed any sign of conducting a "decapitating operation on our headquarters or military attack against our country."

North Korea has launched dozens of missiles this year, several flying over Japan, as it accelerates its program aimed at enabling it to target the United States with a nuclear-tipped missile.

Trading insults

The flight follows a week of heightened rhetoric from Washington and Pyongyang, with Trump and Kim Jong-un trading insults. Trump called the North Korean leader a "madman" on Friday, a day after Kim dubbed him a "mentally deranged U.S. dotard."

Pyongyang conducted its sixth and largest nuclear test on Sept. 3 and has launched dozens of missiles this year. The North has threatened to test a hydrogen bomb over the Pacific.

The Pentagon said the B-1B Lancer bombers came from Guam and the U.S. Air Force F-15C Eagle fighter escorts came from Okinawa, Japan. It said the operation showed the seriousness with which it took North Korea's "reckless behaviour."

Kim Jong Un of North Korea, who is obviously a madman who doesn't mind starving or killing his people, will be tested like never before! —@realDonaldTrump

The patrols came after officials and experts said a small earthquake near North Korea's nuclear test site on Saturday was probably not man-made, easing fears Pyongyang had exploded another nuclear bomb just weeks after its last one.

Tensions have continued to rise around the Korean Peninsula since Pyongyang carried out its sixth nuclear test, prompting a new round of UN sanctions.

Trump told the UN on Tuesday the U.S. would "totally destroy" North Korea if it forced to defend the U.S. or its allies.

North Korea's nuclear tests to date have all been underground, and experts say an atmospheric test, which would be the first since one by China in 1980, would be proof of the success of its weapons program.