The 7 miles/sec number is the number necessary to escape earth's gravitational "well". Think of it this way. If you throw an object up in the air, the faster you throw it, the higher it goes before coming back. 7 miles/sec is how fast you have to throw it so that it never
Now you can also escape earth the way you describe: never actually getting up to 7 miles/sec but firing the rocket continuously over a much longer time. But as admin pointed out, you're better off doing it all at once.
Of course the 7 miles/sec number is from the surface of the earth. In general, it's given by (if I can figure out how to make the image feature work)
where G is the universal gravitational constant (6.67 x 10^-11), M is the mass of the planet (or other round celestial object) in kg, and R is the initial distance from the center of the planet, in meters. (The resulting speed will be in meters per second, not miles per second.)
So if you have a space elevator and can launch your rocket from a much larger R, the speed you need to attain is smaller. Which is good, because the amount of fuel you need goes up exponentially
with the amount of speed (delta v) you need.
"It's only Neutron. We call him that because he's so positive." --from This Island Earth