Show that if at least one of κ > 0 and λ > 0 is infinite, then κ + λ = κλ = max{κ, λ}.
My proof: Assume without loss of generality, κ > λ. If λ = 1, then by definition that at least one is infinite, κ. When λ $\geq$ 2, then κ $\leq$ κ + λ $\leq$ κ + κ $\leq$ 2κ $\leq$ κλ $\leq$ $κ^2$ = κ by the fundamental theorem of cardinal arithmetic.