I would like to show that the supremum of a family $(L_i)_{i\in I}$ of affine functional defined on a topological vector space is convex. Here is my strategy : we will show that the epigraph of $f(x)=\sup_{i\in I}L_i(x)$ is convex.
Consider $(y_1,x_1)$ and $(y_2, x_2)$ in the epigraph of $f$ and $t\in[0,1]$. We have $y_1\geq f(x_1)$ and $y_2\geq f(x_2)$ that is for all $i\in I$
$$ y_1\geq L_{i}(x_1)\quad\text{and}\quad y_2\geq L_{i}(x_2)\implies ty_1 +(1-t)y_2\geq tL_{i}(x_1) + (1-t)L_{i}(x_2)\geq L_{i}(tx_1 + (1-t)x_2) $$
Where the last inequality follows by convexity of affine functional. Since this holds for all $i\in I$ we get : $ty_1 +(1-t)y_2\geq f(tx_1 + (1-t)x_2)$ wich shows that $(ty_1 + (1-t)y_2, tx_1 + (1-t)x_2)$ belongs to the epigraph of $f$ so it is a convex set.
Is this seems correct to you and if not what can I change ? Do not hesitate to provide your insights !