## 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

## Agregation 2013 – Analysis – Part 3

**Part III: Muntz spaces and the Clarkson-Edros Theorem**

Recall that for every we define and that is a strictly increasing sequence of positive integers.

1. Suppose that and . Let . Define and by recurrence for define by

(a) Calculate and prove that

(b) Prove that for every is a linear combination of .

(c) Prove that for every we have .

(d) Deduce that .

(e) Conclude that .

From here on suppose that is arbitrary and the series converges. For denote . For denote the cardinal of the set .

## 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 .

## Increasing sequence of real numbers

Define the sequence in the following way: is a positive integer and

Prove that from a certain moment, the sequence is increasing.

## Interesting sequence

Consider an infinite sequence such that for any positive integers. Prove that if for all , then .

*IMO Shortlist 2002*

## 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.

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