I was just trying to solve the problem from a book by David M. Burton.
If any help in formulating a solution can be rendered, I shall be highly obliged.
thanks
kaka
Prove this by induction. I'll leave the base case of $\displaystyle n=4 $ to you.
Now suppose $\displaystyle p_n<p_1+p_2+p_3+\ldots+p_{n-1} $, then $\displaystyle p_n+p_n<p_1+p_2+p_3+\ldots+p_{n-1}+p_n $.
But Bertrand's Postulate tells us that $\displaystyle p_{n+1}<2p_n $.
So we get $\displaystyle p_{n+1}<2p_n<p_1+p_2+p_3+\ldots+p_n $, which proves the claim.