Questions attached in picture, involves real analysis 3 a) For any natural number n, let s(n) be the smallest natural number m so that 7^m n.

Questions attached in picture, involves real analysis 

3 a) For any natural number n, let s(n) be the smallest natural number m so that 7^m > n. Determine and prove whether the series diverges or converges. 

b) With s(n) as above, determine whether the series converges or diverges.

4. a) Let f be a continuous function on [0, 1]. Show f(c) = A. 

b) Show that the equation c^6 -  1 = 700,000 (c-1) has no solutions with 1 < c < 10 

5. a) Let t(x) be a function which is continuous and differentiable on the entire real line. 

b) With f(x) as above, show that f'(x) is not continuous at x = 0.