linear systems - Impulse Response to LTI

I am new to DSP, and I am self-studying using mostly Proakis. I have a question. There are some examples in the text where you will be given the impulse response of an LTI system, and then asked to solve something/prove something, so forth. My confusion involves the following: How do I know that there exists an LTI system with the given impulse response? For instance, if a problem is posed like one of the examples in Chapter 2, "given a relaxed LTI system with impulse response:

$$h[n] = a^n u[n]$$

where $-1 < a < 1$," (and $u[n]$ is the discrete-time unit step function) how do I know that this impulse response actually comes from an LTI system? Sorry if this question is simple. I'm probably not seeing something.