prove that a non constant periodic, continuous function has a "smallest period"
Let $P$ be the set of periods of $f$. Using your definition, $P$ is non-empty and bounded below by $0$. Consider $p^*=\inf P$. Take $p_n \in P \to p^*$. Fix $x \in \mathbb R$. Then $x+p_n \to x+p^*$ and $f(x+p^*)= \lim f(x+p_n)=f(x)$.
If $p^*>0$, then $p^* \in P$ and so $p^* = \min P$.
If $p^*=0$, then we need to argue that $f$ is constant.
For a more conceptual approach, here is a roadmap:
The set of periods of a function is an additive subgroup of $\mathbb R$.
An additive subgroup of $\mathbb R$ is either cyclic or dense.
The set of periods of a continuous function is a closed set.
A continuous function with a dense set of periods is constant.
Outline of the proof: Assume by contradiction that there is no smallest period. Use first the fact that the difference between two periods is also a period to show that you can find a decreasing sequence of periods which converge to 0.
This means that for each $\delta >0$ you can find some $T$ period such that
$$0 < T <\delta$$
Now, pick $x,y$ arbitrary.
Fix $\epsilon >0$, then there exists a $\delta$ such that for all $z$ with $$|y-z| < \delta \Rightarrow |f(y)-f(z)|<\epsilon$$
Pick some $0< T < \delta$.
Show now that there exists some $n \in \mathbb Z$ such that $|(x+nT)-y|<\delta$.
Then $$|f(x)-f(y)|=|f(x+nT)-f(y)| <\epsilon$$
Since this is true for all $\epsilon$ we get $f(x)=f(y)$. As those are arbitrary, you are done.