1. 14. Let f : D → R and let c be an accumulation point of D. Suppose that a ≤ f (x) ≤ b for all x ∈ D with x ≠ c, and suppose that lim x → c f (x) = L. Prove that a ≤ L ≤ b. 2. Let f and g be...

1. 14. Let f : D → R and let c be an accumulation point of D. Suppose that a ≤ f (x) ≤ b for all x ∈ D with x ≠ c, and suppose that lim_{x → c}
f (x) = L. Prove that a ≤ L ≤ b.

2. Let f and g be functions from D into R and let c be an accumulation point of D. Suppose that there exist a neighborhood U of c and a real number M such that | g (x) | ≤ M for all x ∈ U ∩ D. If lim_{x → c}
f (x) = 0, prove that lim_{x → c}
( f g) (x) = 0.