While digging in MSE I found the following
Linear combination of periodic functions need not to be periodic. If $f_1, f_2, f_3, \dots, f_k$ are periodic functions with fundamental periods $T_1, T_2, T_3, \dots, T_k$ respectively, then $\sum _{i=1} ^k a_i f_i$ where $a_1, a_2, \dots, a_k$ are constants is periodic if $\text{lcm } (T_1, T_2, \dots, T_k)$ exists.
Does anyone has a proof available or a link to it?