Proving that every trap is a lure I am using the book "Learn Limits Through Problems". It states that an interval on a infinite sequence is a "trap" if a finite terms lie outside the interval while an interval will be called a "lure" if an infinite amount of terms lie within the interval. If I'm using the harmonic series as an example, I would think that the whole interval would be considered a trap since there are finite amount of terms (0) outside the interval. It also makes sense that this whole interval would be considered a lure since there are infinite points within the sequence. However, I am not sure how to formally prove this. Any hints will be appreciated. Thanks.

Suppose I have infinitely many beads (terms in the sequence) and a basket (interval). I put some beads in the basket and leave some outside. If there are only finitely many beads left outside (basket is trap) then how many must have been put inside?