Let $V$ be the set of sequences whose terms are contained in $\mathbb{R}^n . V$ is the set of functions $x(·) : N → \mathbb{R}^n $ which we denote as $\{x_n\}_n \subset \mathbb{R}^n$. $V$ is a vector space. $V_0 \subset V$ consists of all convergent sequences, ie:
$V_0 := \{\{x_n \}_n ∈ V : \exists \space \space x ∈ \mathbb{R}^n | x = lim_{n→∞} x_n \}$
Let $A\subset \mathbb{R}^n$ and $V(A):= \{\{x_n\}_n \in V_0 : lim_{n→∞}x_n \in A\}$
Prove that if $A$ is convex, then $V(A)$ is convex
I am struggling to prove this, and also cannot interpret what is meant by $\{\{x_n\}_n$ ? and what $V(A)$ is.
Is $V(A)$ simply the set of the convergent sequences $\{x_n\}_n$ which converge in the set $A$?
Hint: if $x_n \to x$ and $y_n \to y$, what about $t x_n + (1-t) y_n$?