Let . If , prove that .

## An identity

If , prove that

## A finite product

Evaluate the product

**Solution**

We are making use of the identity

Hence,

## A convergent sequence

Let be a strictly increasing sequence of positive integers. Prove that the series , where denotes the least common multiple, converges.

**Solution**

We have successively:

## A limit

Let . We define the sequence . Prove that