Subject: Is time required for a body to attain perfect equilibrium finite or infinite?

i think that heat flows hot body to cold body.if heat is flowing from point A 
to Point X then all subsequent point should have temp.-
temp(A)>temp>temp(X).if a rod is connected to a body maintained a constant 
100cel.then intutively i say that points closer to source will attain 100cel 
first i.e temp variation is= source 100,100,100,99.99,99.98,so on
but this violate the above inequality and flow cannot continue to farther 
pionts without decreasing the temp of closer points(heat cannot flow between 2 
point in equilibrium)thus conclusively i figured that heat dispersion 
decreases as body attains equilibrium and time to attain perfect equilibrium 
tends to reach infinity."But validity of my conclusion is doubtfull".hence i 
wish a expert to explain the situation more clearly.
i am badly in need of the answer as i cannot live freely with a doubt so 
basic...

Re: Is time required for a body to attain perfect equilibrium finite or infinite?