ISI 2015 MMath Entrance Subjective Problems

This post contains problems from Indian Statistical Institute, ISI 2015 MMath Entrance Subjective Problems. Try to solve the problems.

If you know any other problem, please put it in the comment section.

1. Let $ (a_k) $ be a sequence in (0, 1) . Prove that $ a_k to 0 $ if $ \displaystyle{ \frac{1}{n} \sum _{k=1}^n to 0 } $
2. Let $ f: setR $ to $ setR $ be a continuously differentiable function such that $ \infty_{x \in setR} f'(x) > 0 $ . Prove that there exist some $ a \in set R $ such that f(a) = 0.
3. Let X be a metric space such that $ A \subset X $, A is an uncountable subset of X . Prove that X is not separable. Given {d(x, y) : $ x \in A , y \in A $ } > 0
4. Let the differential equation be y'' + P(x) y' + Q(x) y = R(x)  where P, Q, R are continuous functions on [a, b] .