$\displaystyle \textbf{Proof.} \text{ We have that:}$

$\displaystyle \text{i) } d_{n} \in \left [ 0, 1 \right ] \ , \ \forall \ n \in \mathbb{N}$

$\displaystyle \text{ii) } c_{n} = d_{n}a_{n} + (1 - d_{n})b_{n}$

$\displaystyle \text{iii) } \limsup \left \{ a_{n} \right \} < \infty \text{ and } \limsup \left \{ b_{n} \right \} < \infty$

$\displaystyle \text{Note that, } \forall \ n \in \mathbb{N} \ , \ d_{n}a_{n} + (1 - d_{n})b_{n} \text{ is a convex combination between } a_{n} \text{ and } b_{n} \text{. Hence:}$

$\displaystyle (d_{n}a_{n} + (1 - d_{n})b_{n} \leq a_{n}) \text{ or } (d_{n}a_{n} + (1 - d_{n})b_{n} \leq b_{n}) \ , \ \forall \ n \in \mathbb{N}$

$\displaystyle \Rightarrow (c_{n} \leq a_{n}) \text{ or } (c_{n} \leq b_{n}) \ , \ \forall \ n \in \mathbb{N}$

$\displaystyle \Rightarrow (\limsup \left \{ c_{n} \right \} \leq \limsup \left \{ a_{n} \right \} ) \text{ or } (\limsup \left \{ c_{n} \right \} \leq \limsup \left \{ b_{n} \right \} )$

$\displaystyle \Rightarrow \limsup \left \{ c_{n} \right \} \leq \max \left \{ \limsup \left \{ a_{n} \right \}, \limsup \left \{ b_{n} \right \} \right \}$

$\displaystyle \text{Q.E.D.}$