I think I'm almost there but I'm messing up on some step or two...
Suppose G is not connected, therefore G has two or more components such that C1 + C2 + C3+ ... + CM = n
Let Δ belong to C1, therefore C1 has size Δ+1.
C1 + C2 <= n
Δ+1 + C2 <= n
C2 <= n-Δ-1
Δ+1+n-Δ-1 <= n
Δ+1+(n-Δ-2) <= n-1
Δ+1+δ <= n-1
Δ+δ <= n-2
Δ+δ < n-1
Therefore if δ+Δ >= n-1, G is connected
Any tips, I think this proof looks a little weak and I'm pretty sure I'm messing up on the δ. For a disconnected graph I'm pretty sure that δ=n-Δ-#of components, but I'm not sure if I'm allowed to just put that in.