Let us have an one-variable convex function $f: \mathbb{R} \to \mathbb{R}$. Given a strictly monotone sequence $(a_n)_{n \in \mathbb{N}}$ on $\mathbb{R}$. Form the piecewise linear function from the corresponding points on the graph by connecting them in the correct order (we connect $(a_i,f(a_i))$ and $(a_{i+1},f(a_{i+1}))$ for all $i$ from $1$.
Is it necessary that the newly-formed piecewise linear function must be a convex function? This question has been bugging me a lot.
I can tell that the answer is highly to be positive, from some sketches I draw. But somehow the proof is not obvious to me.
Did I make a mistake, or I overlooked something? Please give me a hint.
Thank you for reading.
EDIT: I forgot to put $f$ as a convex function. The answer would be obvious otherwise.