Those little plastic or metal tubes on the end of shoelaces are called aglets.
One of mine fell off the end of my lace. Neither the aglet nor the lace are damaged, but it's likely impossible to just push the aglet back on, and I spent a while trying, because the lace's weave loosened a bit.
My aglet is a solid plastic tube, so I can't pry it open then reclose it. How can I get this thing back on the lace?