Why does autocorrelation function for trend in data decrease with increasing lag? Forecasting: Principles and Practice, 3rd edition by Hyndman and Athanasopoulos states in section 2.8, Autocorrelation that, for trend in data the autocorrelation function has positive values that slowly decrease as lag increases.
The definition for autocorrelation function for lag $k$ is defined here as -
$r_k=\frac{\sum_{t=k+1}^{T}{(y_t-\overline y)(y_{t-k}-\overline y)}}{\sum_{t=1}^T (y_t-\overline y)^2}$, where $T$ is the length of the time-series.
Can anyone give a rigorous proof and intuitive sense as to why it might be so, indifferent of the type of trend that the series might have?
P.S.: I am new to time-series; however, comfortable with maths.

Here is the full quote from the section you reference (bold added):

When data have a trend, the autocorrelations for small lags tend to be large and positive because observations nearby in time are also nearby in value. So the ACF of a trended time series tends to have positive values that slowly decrease as the lags increase.

Surely that gives the intuition. As the lags increase, the values are further apart in both time and value due to the trend.
