Cosets and subgroup

Let $G$ be a group and $H$ be a subgroup of $G$. Let $b \in H$. Prove that:

$$Hb =H$$

where $Hb$ is the left coset of $H$.

Solution

We are first proving that $Hb \subseteq H$. From the definition of $Hb$ we have that:

$$Hb=\left \{ hb \mid h \in H \right \}$$

However since $b, h \in H$ we also have that $hb \in H$. Thus $Hb \subseteq H$.

Now, we are proving that $Hb \supseteq H$. Since $h \in H$ and $\underbrace{hb^{-1}}_{h_1}b=h_1 b$ the result follows.