Taking the limit of voltage division equation
As Andy aka points out, it's not entirely clear what you're asking. A schematic would help in the future. I think your original equation is incorrect. It sounds like you have this circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

From the basic voltage divider equation:
$$
V_{out}=\frac{R_LV_s}{R_L+R_2}
$$

Where RL is R1 || RM.

$$
R_L=\frac{1}{\frac{1}{R_1}+\frac{1}{R_M}}
$$

Showing that RL -> R1 as RM -> infinity is simple algebra from this point.

Now, as a matter of reality, RM is not infinite. This approximation only holds value if RM is much larger than R1. If RM is within, say, two orders of magnitude of the value of R1, the meter can affect the circuit it's attached to.