(8 pts.) a. Determine limsup_{n \to \infty} a_{n} and liminf_{n \to \infty} a_{n}.

(9 pts.) b. Determine any other subsequential limits of \{ a_{n} \}.

(17 pts.) 2. Use an \epsilon-\delta argument to prove that lim_{x \to 2} {x+2 \over x+3} = {4 \over 5}.

(17 pts.) 3. Use an \epsilon-\delta argument to prove that lim_{x \to {1/2}} \ { 1 \over x} = 2.

(17 pts.) 4. Let a_{n} and b_{n} be bounded. Prove that limsup_{n \to \infty} ( a_{n} + b_{n} ) \leq limsup_{n \to \infty} a_{n} + limsup_{n \to \infty} b_{n}.

(16 pts.) 5. Let S be a bounded set of real numbers that contains an infinite number of distinct points. Prove that the set S has an accumulation point.

(16 pts.) 6. Let a_{n} be a bounded sequence. Let limsup_{n \to \infty} a_{n} = L. Show:

(1) for every \epsilon >0 there exists N such that if n>N then a_{n} < L + \epsilon and

(2) for every \epsilon >0 and for every N there is an n_{0}>N such that a_{n_{0}} > L - \epsilon.

BONUS! (7 pts.) 7. Let a_{n} be a Cauchy sequence. Show that a_{n} is bounded.

BONUS! (5 pts.) 8. Give an example to show that the inequality in problem 4 may be a strict inequality.