Let $(d_1, d_2, \dots, d_n)$ with $0 \leq d_1 \leq d_2\leq \cdots \leq d_n$ be a degree sequence of a graph $G$ with $n$ vertices. Show that if $d_j \geq j+k$ for all $j \leq n-1-d_{n-k}$. Then $G$ is $k+1$-connected.
This is a homework problem, it seems that it is related to a theorem by Bondy, 1969 or Boesch, 1974. But I can't figure it out myself and can't find a proof for it.
Can anyone help?