3. Prove that a sequence {a,} converges to the real number a if and only if lim sup a, = lim inf a, = a. (HINTS: For the direction, you'll need the two Lemmas given in Session 25. For the direction,...

Extracted text: 3. Prove that a sequence {a,} converges to the real number a if and only if lim sup a, = lim inf a, = a. (HINTS: For the direction, you'll need the two Lemmas given in Session 25. For the direction, the Squeeze Theorem is quite helpful.) Lemma: Every seguence {an} has a Subsequence {any} that converges to limsup an. Lemma: Every sequence žans has a subsequene {anps that converges to liming an.