elem#theo - induction + Bertrand's

OK: $\displaystyle p_{n}$ denotes the $\displaystyle nth$ prime. For $\displaystyle n>3$, show that $\displaystyle p_{n} < p_{1} + p_{2} + ... + p_{n-1}$

the hint was to use induction and Bertrand's conjecture

ok: so the answer is in the book, but its not helpful b/c I don't get one jump

so they do this:

So for n=4, $\displaystyle p_{4} = 7 < 2+3+5$, which is true.

Assume for some k, $\displaystyle p_{k} < p_{1} + p_{2} + ... + p_{k-1}$

By Bertrand, there exists a prime between $\displaystyle p_{k}$ and $\displaystyle 2p_{k}$

Here is where I'm lost: they go from there to $\displaystyle p_{k+1} \leq 2p_{k} < (p_{1} + p_{2} + ... + p_{k_{1}})$,

thus proving that statement is true for $\displaystyle k+1$

How? *so confused*