from (f) we have:

x/p_{j+1} + x/p_{j+2} + ...\ge x-N_j(x),

form (d) we have:

N_j (x)< x/2.

So:

x/p_{j+1} + x/p_{j+2} + ...\ge x-N_j(x)>x-x/2=x/2

Now divide through by x:

1/p_{j+1} + 1/p_{j+2} + ...>1/2,

which from (e) we know means that:

1/1 + 1/3 + ..+ 1/p_n + ...

diverges.

(sorry but LaTeX seems to be not working again!!) I think I have sorted out the LaTeX now, but will leave this here just in case.

RonL