How can I prove that for a natural number K, a language that accepted by a Turing machine with K cells after the input word ends, belongs to R (which R is the set of languages that there is Turing machine that accept them while for each input the run is finite)?
Given the length of the input $n$, you can come up with a bound $N(n)$ on the number of configurations that the Turing machine can have. If the machine hasn't stopped on an input within $N(n)$ time steps, then it will never stop (why?), and you can use this to complete the proof. Details left to you.