Understanding the Conditions for Recovering a Discrete Time Signal Through Sampling
Lets have a higher level of the idea of signal reconstruction from samples.

When you try to reconstruct something from partial information it is important to know what you know about the result required.

For instance, if you know the signal is a Line if $ \mathbb{R}^{2} $ then all you need is 2 samples, any samples of it.
You can even generalize that to any polynomial with having enough points samples from it.

You can even find more interesting rules (For instance, can you think about the case of having single Sine Function with Zero Phase ad Unit Amplitude, how many point will you need?).

So when dealing with reconstruction we need assumption of the model and then derive the requirements from the samples.  

If we assume the signal has limited bandwidth, namely from a given frequency and above it vanishes in the Fourier domain then the Nyquist Shannon Sampling Theorem states a simple rule, sample the signal in a uniform manner with sampling rate which is larger then twice the bandwidth of the signal.
Once you do that, you have enough information to reconstruct it.

So in your case, saying that the signal vanishes from certain frequency and above is just stating it obeys the assumption of the model.