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Is this true statement, “difference in finding the longest path from two graphs $G_1$ and $G_2$ is bounded by $O(nm)$”

Let two weighted graphs $G_1$, $G_2$ such that $\vert G_1\vert=n, \vert G_2\vert=m$. Let $d_1$ be the length of the longest path in graph $G_1$ and $d_2$ be the length of the longest path in $G_2$. Let $d=\max(d_1,d_2)$.
The statement is,
$$\vert d(G_1,G_2)-d\vert \le O(nm).$$
I have problems to prove this statement. Can anyone help me in this problem?

A:

Hint:
Show that the longest path in a graph $G$ is of length $O(n\sqrt{V(G)})$.
Proof:
Since $G$ has at least one node, we can assume $n\geq 1$. Consider the definition of $d$. If $d$ is the length of the longest path in $G$, then $d=\max\{d_1,d_2\}\geq\min\{d_1,d_2\}=d_1\ve

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