 Solution

Let $b_{\ast }$ be the smallest threshold for which the edges of length at most $b_{\ast }$ connect $x$ to $y$. This is exactly the optimum value of the bottleneck shortest path problem. While any edge longer than $b_{\ast }$ is being processed, every edge of some threshold-$b_{\ast }$ route is still present, because shorter edges have not yet been processed. Thus all edges longer than $b_{\ast }$ are deleted. The final path from part (i) consequently has maximum edge length at most $b_{\ast }$. It cannot have a smaller maximum by the definition of $b_{\ast }$, proving optimality. Ties in edge ordering do not affect this argument.