---
tags: teaching
---

[练习题](/PUD0T_3KQZy2FWJ2nZ6V_A)
### Excercise of writing a proof.

**Ex** Show that
$$
\{x:x=4m\text{ for some }m\in \mathbb Z\}\subset\{y:y=2n\text{ for some }n\in\mathbb Z\}
$$

**Proof:**

For any $a\in\{x:x=4m\text{ for some }m\in \mathbb Z\}$, we have
$$
a=4m\text{ for some }m\in\mathbb Z.
$$
Let $n=2m$ then we can write
$$
a=2(2m)=2n
$$
this implies $a\in \{y:y=2n\text{ for some }n\in\mathbb Z\}$.

Therefore, we proved that
$$
\{x:x=4m\text{ for some }m\in \mathbb Z\}\subset\{y:y=2n\text{ for some }n\in\mathbb Z\}
$$