By the Definition of Orlicz function $M$, we know that $M(0)=0$, $M(x)>0$ for $x>0$ and $M(x)\rightarrow \infty$ as $x\rightarrow \infty$. And the Orlicz-Luxemborg norm is given by $$\|x\|=\inf\left\{r>0:\sum_{n=1}^{\infty}M\left(\frac{x_n}{r}\right)\leq 1\right\}.$$
Using this, can we conclude that $\sum_{n=1}^{\infty}M\left(\frac{x_n}{r}\right)\leq 1$ implies $x_n$ is a bounded sequence.
If $(x_n)$ was not bounded, then there would be a subsequence $\left(x_{n_k}\right)_{k\geqslant 1}$ such that $\lvert x_{n_k}\rvert\gt k$ for all $k$. Therefore, $$ 1\geqslant\sum_{n=1}^{+\infty}M\left(\frac{\lvert x_{n}\rvert}r\right)\geqslant \sum_{k=1}^{+\infty}M\left(\frac{\lvert x_{n_k}\rvert}r\right)\geqslant \sum_{k=1}^{+\infty}M\left(\frac{ k }r\right) $$ and since $M(x)\to +\infty$ as $x$ goes to $+\infty$, we get a contradiction.