Proof that sequence where two entries are the same, then the gcd of these indices must be the step

Question

I am trying to prove the following. Assume there is a bit string B = $\{0,1\}^n$, and given that $ B_i = B_{i+t} $ and $ B_i = B_{i+s} $ for all i < n. then it must hold that $B_i = B_{i+p} $ where p = gcd(s,t)

I cant go any further than this: $ B_i = B_{i+t} $, $ B_i = B_{i+s} $ then $ B_i+t = B_{i+s} $ then $ B_{s} = B_{t} $