Set of positive bounded numbers in 2 dimensions, excluding zero

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I want to state that a robot needs to be inside a bounded area. I would like to have a nice short expression like the one below:

$$x \in \mathbb{R}_+^2$$

But what I actually mean is the following:

$$ x_1 \in (0,x_{lim}] \\ x_2 \in (0,y_{lim}] $$

Which looks rather bulky, and is not easily generalized to higher dimensions. Is there any compact way of stating this?