§ Low pass filter by delaying

I don't understand this. Full text with context:
For example, at a sample rate of 10kHz, one sample delay is equal to the sample time of 0.0001 seconds. ... A signal at 1000Hz has a period of 0.001 seconds and a one sample time delay is 1/10 of the period, or 36°.
I understand the math upto this point. 0.001 seconds -> 360 degrees, so 0.0001 seconds -> 36 degrees. Thus, a sample delay of 0.0001 seconds corresponds to 36 degrees of phase shift. What's the next bit? I don't understand this:
The larger phase difference results in a lower amplitude and the higher frequency signal is attenuated more than the lower frequency signal. The effect is that of a low-pass filter.
  1. Why does a large phase shift result in a lower amplitude?
  2. Why is the higher frequence signal attenuated (= weakened, as far as I understand) more?
Wiki claims that this discrete process is the "same as" the low pass filter: 
function lowpass(real[0..n] x, real dt, real RC)
    var real[0..n] y
    var real α := dt / (RC + dt)
    y[0] := α * x[0]
    for i from 1 to n
        y[i] := α * x[i] + (1-α) * y[i-1]
    return y
One proof strategy that I can think of:
  1. Split signal into fourier components
  2. Notice that if the frequency is higher than some threshold, we will start introducing phase deltas larger than π\pi. This will cause cancellation of a sin⁡\sin wave.
  3. See that this makes sure that only low frequency signals live.
I'm now trying to understand why this doesn't "wrap around". I would naively expect: (1) low frequency = minimal destructive interference (2) high frequency (phase difference 180 degrees) = maximal destructive interference (3) EVEN HIGHER frequency (phase difference greater than 180 degrees) = non-maximal destructive interference? I believe the resolution is to remember that we only build signals upto what Niquist lets us, so we build at max a sin wave of frequency fs/2 where the same period is fs. Thus, when we phase shift, we will get a phase of at max 180 degrees. As to why the Niquist ratio is 1/2, imagine a circle. a 1/2 is how much "ambiguity" we have in the phase. In that, if we jump by more that pi, (ie, more than 1/2 of the circle), we will ????