Problem: Is set connected and compact

Question

Is $ \left\{ (x,y, 1+x+y) \in \mathbb{R^3} \mid x,y \in [1,2] \cap \mathbb{I} \right\} $ connected as a subspace of $(\mathbb{I^3} , d_{2} )$? I guess it isn't connected since it is a subset of $\mathbb{I^3}$ which isn't connected. But is it compact?

Answer 1

I think it is not compact because the sequence $(1 + \frac{1}{\sqrt n},1 + \frac{1}{\sqrt n}, 3+ \frac{2}{\sqrt n})$ has a limit $(1,1,3)$ which isn't in the set.