I.S.I. 2015 M.Math Entrance Subjective Problems

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

  1. Let \((a_k) \) be a sequence in (0, 1) . Prove that \(a_k to 0 \) iff \(\displaystyle{ \frac{1}{n} \sum _{k=1}^n to 0 } \)
  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] .

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