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Determine if this space is complete
Let $T_r = {(x_1,…,x_n) in mathbb{R}^n: sum_{i=1}^n x_i = r}$ and $T_infty = {(x_1,…,x_n) in mathbb{R}^n: x_i in mathbb{R} forall i}$. Find the dimension of $T_r$ and $T_infty$
The answer I have is $T_infty = mathbb{R}^n$ and $T_r$ has dimension $r-1$
I have to solve it by finding a subset of $T_infty$ that is dense and show that this subset is uncountable. What I tried is, $T_infty$ is complete and $T_r$ is a closed and bounded subset, so $T_r$ is complete. How would I show that this set is dense?
A:
The set is dense. Let ${(x_n}_nsubset T_r$ and $xin T_infty$. Then $x_nto x$ as $
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