Let  { 𝑎 𝑛 } 𝑛 = 0 ∞ , { 𝑏 𝑛 } 𝑛 = 0 ∞   and  { 𝑐 𝑛 } 𝑛 = 0 ∞   be positive sequences that are divergent to ∞ . Assume that 𝑎 𝑛 << 𝑏 𝑛 . Which of the following statements must be true? Select all the correct answers.多项选择题

IF ∀ 𝑛 ∈ 𝑁 ,     𝑏 𝑛 = 2 𝑐 𝑛 ,  THEN 𝑎 𝑛 << 𝑐 𝑛 .

IF 𝑏 𝑛 << 𝑐 𝑛 ,  THEN 𝑎 𝑛 << 𝑐 𝑛   .

IF lim 𝑛 → ∞ 𝑐 𝑛 𝑏 𝑛 = 0 ,  THEN lim 𝑛 → ∞ 𝑎 𝑛 𝑐 𝑛 = 0 .

lim 𝑛 → ∞ 𝑏 𝑛 𝑎 𝑛  DNE

IF ∀ 𝑛 ∈ 𝑁 ,     𝑐 𝑛 = 𝑎 𝑛 + 𝑏 𝑛 2   ,  THEN 𝑎 𝑛 << 𝑐 𝑛 .

IF ∀ 𝑛 ∈ 𝑁 ,   𝑐 𝑛 = min { 𝑎 𝑛 , 𝑏 𝑛 }   ,  THEN ∀ 𝑛 ∈ 𝑁 ,   𝑐 𝑛 = 𝑎 𝑛  .