若 ∃N∈N,∀n>N,zn≤xn≤yn\exists N \in \mathbb N, \forall n > N, z_n \leq x_n \leq y_n∃N∈N,∀n>N,zn≤xn≤yn, 且 limn→∞zn=limn→∞yn=A\lim_{n \to \infty} z_n = \lim_{n \to \infty} y_n = Alimn→∞zn=limn→∞yn=A ⟹ limn→∞xn=A\implies \lim_{n \to \infty} x_n = A⟹limn→∞xn=A
若 ∀x∈U∘(a,δ),g(x)≤f(x)≤h(x)\forall x \in \mathop{U}\limits^\circ (a, \delta), g(x) \leq f(x) \leq h(x)∀x∈U∘(a,δ),g(x)≤f(x)≤h(x), 且 limx→ag(x)=limx→ah(x)=A\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = Alimx→ag(x)=limx→ah(x)=A ⟹ limx→af(x)=A\implies \lim_{x \to a} f(x) = A⟹limx→af(x)=A
