confusion about telling wether or not a subset is a subspace

Is the following subset of $\displaystyle R^3$ a subspace $\displaystyle R^3$? The set of all vectors of the form (a,b,2)

The answer key has:

W is not a subspace. To show this let $\displaystyle \vec u=(a_1, b_1, c_1)$ and $\displaystyle \vec v = (a_2, b_2, c_2)$ be in W. Then $\displaystyle c_1=c_2=2$ Now $\displaystyle \vec u + \vec v = (a_1+a_2, b_1+b_2, c_1+c_2)=(a_1,b_1+b_1,4)$ which is not in W

Two things confuse me. 1) Why is it $\displaystyle b_1+b_1$ not $\displaystyle b_1+b_2$? 2)I don't see how the property "$\displaystyle \vec u + \vec v$ is in V" fails to hold because $\displaystyle (a_1+b_1+b_1,4)$ is in 3 space isn't it?