Take the map $\gamma: (1,\infty) \to \mathbb{R}^2$ be defined by $t \mapsto (\frac{1}{t}\cos(2\pi t),\frac{1}{t}\sin(2\pi t))$. I'm able to show that it is an injective smooth immersion. Now, I'd like to show that it is a smooth embedding. To this regard, I have to show that $\gamma: (1,\infty) \to \gamma((1,\infty))$ is a homeomorphism.
Well, I suppose that it is sufficient to show that the above map is open or closed. I have no clue. In fact, taking an open set $U$ of $(1,\infty)$, how to describe its image? Or at least, how to see that its image contains an open set?
However, in Guillemin & Pollack's book I found that it is sufficient to show that the map is proper, i.e. the preimage of compact sets is again a compact set. Hence I take a compact $K \subseteq \gamma((1,\infty))$. Then $K$ is compact in $\mathbb{R}^2$, so that it is closed and bounded. What about its preimages? It is quite difficult to work with the function $\gamma$!
This map is unfortunately not proper: indeed, the preimage of the compact set $\{(x,y) \mid x^2+y^2 \leqslant 1 \}$ is $(1,+\infty)$, which is not compact.
However, if $\Gamma = \gamma\left((1,+\infty)\right)$ is the image of $\gamma$, note that the map $$ p \in \Gamma \mapsto \frac{1}{\|p\|} \in (1,+\infty) $$ is a continuous inverse for $\gamma\colon (1,+\infty) \to \Gamma$. This should be sufficient to show that $\gamma$ is an embedding.