Romanian Mathematical Olympiad

Assume, by way of contradiction, that $M$ contains the infinite arithmetic sequence $\{b_1^k, b_2^k, b_3^k, \dots\}$, where $1 \le b_1 < b_2 < b_3 < \dots$.
If we denote by $r$ its common difference, we have $b_{n+1}^k - b_n^k = r$, for all $n \ge 1$.
Observe that
$$r = b_{n+1}^k - b_n^k = (b_{n+1} - b_n) (b_{n+1}^{k-1} + b_{n+1}^{k-2}b_n + \dots + b_{n+1}b_n^{k-2} + b_n^{k-1}).$$
The expression in the second parenthesis is clearly increasing, hence the sequence $b_{n+1} - b_n$ is decreasing, and since all its terms are positive integers, we reached a contradiction.