Re: how can i prove the the clausius inequality?

One doesn't really "prove" the Clausius inequality or any other statement 
of the second law of thermodynamics.  Same goes for the first law.  It is 
just a matter of observing in nature that heat always flows from hot to 
cold, that you never get 100% efficiency in a process, etc.  Once we 
decide from nature that this is "true" with confidence, then we adopt it 
as a basic postulate and move forward from there.

There are actually "proofs" of the Second Law that come out of statistical 
physics, but these proofs (the first was due to Boltzmann) have their own 
assumptions buried inside so one isn't really proving the Second Law from 
first principles, though perhaps one is starting from principles that are 
somewhat more basic.  But for all practical purposes you can think of the 
Second Law as an unproved assumption, but one backed up by lots of 
observation of nature.

One can ask the related question about "proving" a particular statement of 
the Second Law given another statement.  And you did ask that in the other 
message you sent, so I'll deal with that when I get around to replying to 
that (by the way, in the future when you have two almost identical 
questions to ask it would be more polite to combine them into one message).