Hello, My instructor, whilst trying to prove that liminf of sequence a_n = limsup of sequence a_n = A, _ wrote that since we know that a_n0-ε<an<a_n0+ε → a_n0-ε ≤ A ≤ A ≤ a_n0+ε. Why is that true? I mean, how do we know that if a sequence is bounded then its lim inf (i.e. the lowest amongst the limits of its subsequences) and lim sup (the highest amongst the limits of its subsequences; do forgive me for potential misnomers) are indeed within that area bounded by a_n0-ε and a_n0+ε? I hope you may assit in clarifying this. Thank you!