How deep bore well pumps works? From highschool physics that I learned.
Let

density of water be $\rho = 10^3  
 kg/m^3 $
and acceleration due to $g = 10 m/s^2 $
and atmospheric pressure be $P{_0} = 10^5 Pa$

now as $\rho \times g \times h = P{_0}
\implies h = 10 meter $
so one cant suck water more than 10 meter .
now my question is then how deep borewell suck water from really deep sites?
also how do mountaineers get water from sea on ground ?
 A: The deep well pumps do not suck water.  They are submersible pumps that are placed below water level deep inside the well.  Due to that location, they already have a positive suction gauge pressure, and they must have enough discharge pressure to get water up to at least ground level.
A: Pumps only push : atmospheric pressure pushes water to a maximum of about 32 ft ( you can call that 10 meters),but likely not practical for use beyond about 20 ft  of elevation. Deep oil wells use ESP ( electric submersible pumps) and rod pumps. Rod pump is in the bottom of the well and up to two miles of steel ( sucker) rods go to the pump to operate it. ESP are centrifugal pumps (in the bottom), normally with many stages, maximum of more than 50 stages (impellers). As the bulk of what an oil well produces is water , the pumps can easily pump all water.
A: 
how do mountaineers get water from sea on ground?

I'm not sure exactly what you mean here because I don't think mountaineers have much use for sea water.  Most of the water on land (including mountains) was evaporated from the oceans and then falls as precipitation and collects into streams and lakes.  The heat from the sun provides the energy needed for that to occur.  I expect most mountaineers bring the water they drink with them on their journey.
