Is $\Bbb R$ a convex set?

Question

I was looking for some proofs that $\Bbb R$ is a convex set, but I could not find any. So I have calculated some results of a loss function and these results are, let us say, a subset of real numbers set. Is the set of real number $\Bbb R$ a convex set? Could you please provide any proof? and if any subset of $\Bbb R$ is also a convex set?

Answer 1

$\Bbb R$ is convex because it is a subset of the $\Bbb R$-vector space $\Bbb R^1$ and, for all $x,y\in\Bbb R$ and for all $t\in[0,1]$, $tx+(1-t)y\in\Bbb R$.

A subset $S\subseteq\Bbb R$ is convex if and only if it is an interval, in the sense that for all $u,v\in S$ such that $u<v$ and for all $y\in\Bbb R$ such that $u<y<v$, $y\in S$.

Answer 2

A set $X$ is convex if for any $x_1, x_2 \in X$, $\lambda x_1 + (1-\lambda) x_2 \in X$, $\lambda \in (0,1)$.

If you take any two real numbers and one in $(0,1)$ and compute the convex combination, you get a real number. So $\mathbb{R}$ is a convex set.