### Small Steps

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

# Eyes Down

### Why do this problem?

The problem gives practice in working with linear inequalities and in working systematically through separate cases.

### Possible approach

Encourage the class to try some numerical values for $x$, to compare the values of the three functions and to record their findings. Collect sufficient results from the class to provide evidence for spotting patterns and making conjectures.

### Key questions

If the integer part of $x$ is $a$ then $x=a + b$ where $a$ is a whole number and $0\leq b < 1$. What is the difference between the separate cases where $0 \leq b < {1\over 2}$ and ${1\over 2}\leq b < 1$?