Show that $t_n=\min\{m/n \mid m\in\mathbb N, m/n\ge t\}$ converges to $t$

Question

Let $t \in \mathbb{R^+}$. Is it true that $$t_n=\min\{m/n \mid m\in\mathbb N, \ m/n\ge t\}$$ converges to $t$? I feel like it's true, but I have no idea how to show it.

Answer 1

$t_n\ge t$ and also $t_n\lt t+\frac{1}{n}$ (because $t_n-\frac{1}{n}\lt t$) thus the conclusion comes from the squeeze theorem.