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 ${1\over 2}\leq b < 1$?