# Suppose that #f:RR->RR# has the properties #(a)# #|f(x)| le 1, forall x in RR# #(b)# #f(x+13/42)+f(x)=f(x+1/6)+f(x+1/7), forall x in RR# Prove that #f# is periodic?

##### 2 Answers

We are given

Substituting in

Adding these equations and canceling the terms which appear on each side, we get

Next, we can substitute

Repeating this process three more times results in the equation

Next we substitute in

Repeating this process five more times results in the equation

The above equation implies that the difference between

See below.

#### Explanation:

Here

We have

so calling

Also considering

making

and now we can write

or

but

so

solving for

so