There are some simple ways to prove that the moon is not actually changing size including photographs of the moon at the horizon and higher up reveal that they are exactly the same size (if taken with the same camera settings).
Another way of demonstrating that the effect is an illusion is to hold a small coin at arm's length with one eye closed, positioning it next to the seemingly large moon. When the Moon is higher in the sky positioning the same coin near the moon will reveal no change in size.
The moon illusion occurs primarily through visual angle illusions. Click here for a more in depth discussion with multiple approaches to explaining this phenomenon.
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