I have the following question:
Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ is linearly independent for some complex point $z\in\mathbb Z$.
Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ is linearly independent for all $z\in\mathbb Z$.
I have the following question:
Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ is linearly independent for some complex point $z\in\mathbb Z$.
Prove that the set $\{\exp(b_{n}z):n\in\mathbb N\}$ is linearly independent for all $z\in\mathbb Z$.
Pending clarification from OP of the matters Hagen raised in the comments, these remarks may be useful.
Suppose $b_n=n$ for all $n$, and take $z=1$, so in the first question we are talking about the numbers $e,e^2,e^3,\dots$. That these numbers are linearly independent over the rationals is another way of saying that $e$ is a transcendental number. The transcendence of $e$ is well-known and not at all trivial to prove.
Given one complex number $z=z_0$ such that the numbers $e^{b_nz}$ are linearly independent over the field of your choice, it is immediate that the functions $e^{b_nz}$ are linearly independent over that field, since any linear combination of the functions will fail to vanish when evaluated at $z=z_0$.