## IMC 2014 Day 1 Problem 2

Consider the following sequence

Find all pairs of positive real numbers such that

**IMC 2014 Day 1 Problem 2**

## SEEMOUS 2014

**Problem 1.** Let be a nonzero natural number and be a function such that . Let be distinct real numbers. If

prove that is not continuous.

**Problem 2.** Consider the sequence given by

Prove that the sequence is convergent and find its limit.

**Problem 3.** Let and such that , where and is the conjugate matrix of .

(a) Show that .

(b) Show that if then .

**Problem 4.** a) Prove that .

b) Find the limit

## Asymptotic characterization in terms of sequence limits

## SEEMOUS 2012 Problem 4

## Nice characterization of convergence

Suppose is a topological space, and consider the sequence with the following property:

- every subsequence has a subsequence converging to .

Then .

## Convex function & limit

Suppose is convex and differentiable with . Prove that . Read more…

## Property of a Sequence

We have bounded a sequence of real numbers such that . Prove that the set of accumulation points of such a sequence is a closed interval.

Read more…