Exercise 5. Consider the sequence (fn)n defined by fn(x) = 1#nz: for r>0. -.... 3. Show that (fn)n does not converge uniformly to f on [0, 00).

3. Let an = (3n +4n)¹/n (a) Prove that for all n € N, 4 < an = 4(1 + (²)n)¹/n < 4.2¹/n (b) Prove that the sequence {an}=1 is convergent and find its limit.

M.

9n Exercise 6. Let (xn) be a sequence with x1 = 1 and xn+1 V6 + xn for n >1. Prove xn → 3. Exercise 7. Let f : E→ R and g: E R where p E E. Prove that if f(x) and g(x) are continuous at x = p, then h(x) = max(f (x), g(x)) is continuous at x = p.