Is it true that it is mostly impossible, for any kind of antibiotic, to succeed in killing off every bacteria in a very rich environment?
Yes. It is down to mutations. DNA replication is not 100% accurate, there will be mistakes that are not corrected in the progeny produced. 

Antibiotics works like a bit of gunk wedging itself into the machine of a bacteria cell but avoids jamming a human cell due to that very same shape. Gunk that wedges into both human and bacteria machines is just a really bad poison. So that gunk has a very special shape.

Resistance can be acquired by recognizing gunk of that particular shape and removing it (not something easy to evolve but very effective when it does) or just change the shape of the machine so the gunk cannot stick. Usually the altered machine doesn't work as efficiently as it can but in an environment polluted by fouling gunk, it works well enough.

And a rich environment means there are many bacteria. 

So if your chances of mutating the right part of the bacteria machine to gain resistance is 1 in million, a rich environment with 10 million cells will have 10 such mutates. 

Of course in reality we are talking about trillions (a million million) of bacteria cells per human (actually 39 trillion bacteria per human) and we are talking about billions of humans.  And we are not even counting the number of bacteria in the soil and in farm animals.  And it takes a few years for resistance to evolve.

When you look at the numbers involved , it is very difficult not to imagine resistances evolving. Even if the chances of that happening is 1 in hundred of trillions. It just takes a single bacteria cell to mutate and so gain resistance for it to spread and pass that resistance to other bacteria.