Why is $e^{x}$ not uniformly continuous on $\mathbb{R}$?

It seems intuitively very clear that $e^{x}$ is not uniformly continuous on $\mathbb{R}$. I'm looking to 'prove' it using $\epsilon$-$\delta$ analysis though. I reason as follows:

Suppose $\epsilon > 0$; in fact, fix it to be $\epsilon=1$.

For contradiction, suppose that $\exists \delta >0$ s.t. $$ (\star) \ |x-y|<\delta \Rightarrow |e^{x}-e^{y}|<\epsilon=1 \text{ for } x,y \in \mathbb{R}.$$Note that $e^{x+\delta}-e^{x}=e^{x}(e^{\delta}-1)$. So, for $x$ large enough (so that RHS $>1$), the relation $(\star)$ does not hold.

This is our contradiction, and so the exponential function is not uniformly continuous on $\mathbb{R}$.

Is this reasoning correct and sufficient?

Thanks.