Showing $\bigcup_{k}\{(a_{1},...,a_{k}):a_{j}\in [0,1]\cap \mathbb{Q} ,\sum^{k} a_{j}=1\}$ is countable Showing  $M=\bigcup_{k}\{(a_{1},...,a_{k}):a_{j}\in [0,1]\cap \mathbb{Q} ,\sum^{k}  a_{j}=1\}$ is countable. 
I find this hard to believe because say for fixed k and any $b,c\in [0,1]\cap \mathbb{Q}$, we have $(\frac{1}{k-2},...,\frac{1}{k-2},b,c)\in M$. So I can embed in M any element of $ P([0,1]\cap \mathbb{Q})$,  which is uncountable.
thanks

Fix $k$. The set $\{(a_1,...,a_k); a_j\in [0,1]\cap \mathbb{Q}, \sum_ja_j=1\}\subset \mathbb{Q}^k$. Therefore it is countable. Countable union of countable sets in countable.
