Let $G$ be a non-complete graph of order $n$ and $k$-connected. Show that if S is a vertex cut of cardinality κ(G), then G−S has at most t components

Question

Let $G$ be a non-complete graph of order n and $k$-connected such that d (v) > $ \frac {n + kt − t} {t + 1} $ for some integer t ≥ 2 and all v ∈ V (G). Show that if S ⊆ V (G) is cut set of cardinality $κ(G)$, then G \ S has at most $t$ components.

I have been trying to solve this problem by contradiction, assuming that $G/S$ has most than $t$ components. However I don't seem getting to a point where I can't arrive to a contradiction with $d(v)$ o with the components.