Velocity problem 
Consider a body thrown up with some arbitrary initial velocity. It can be proven that the speed of the body one second before it reaches its maximum height is  always the same and is independent of the initial velocity of the body. 

How, or rather why is this so? What can be a simple and logical explanation for this?
As a special case, it is obvious that the above is not true for any arbitrary instant of the motion. When does it 'start' becoming true, ie is it valid for 2,3... seconds before the body reaches its maximum height?
Please excuse me for any wrong concepts in my question. I'm new to physics and am really interested!
 A: Work backwards from the object at maximum height when its vertical velocity is zero.  
Its speed at time $t$ before or after reaching that height is $gt$ where $g$ is the gravitational field strength.
The only difference will be that the velocity before reaching maximum height will be upwards and the velocity will be downwards after reaching maximum height.  
The speed at a time one second before reaching maximum height is thus only dependent on $g$.
A: As many comments pointed out, this is true because the acceleration is the same. You say: "it is obvious that the above is not true for any arbitrary instant of the motion".
No, it is true, but, if the initial velocity is different, it will take a different number of seconds to reach the top. 
So, it is true, until one of the two objects you are comparing, going back in time, reaches the time it was launched up in the air. You cannot go back further, but you can for the second object, that was launched at higher speed.
As an aside, if one object was launched at such low speed that it takes less than a second to reach maximum height, that statement is wrong. The authors take one second as a small enough period of time. It could be 1 minute, but then you must assume both objects were launched at initial speeds so high that they take at least one minute to reach maximum height.
