Why every non-discrete locally compact group contains a nontrivial convergent sequence?
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This is true. The difficulty is the case in which the group is not metrizable: Pavel M (and Henno Brandsma) noted that if $G$ is a metrizable locally compact group without isolated points then every neighborhood $U$ of the neutral element is infinite, so it suffices to choose an infinite sequence in a compact $U$ and extract a convergent subsequence.
In the general case, I doubt that there is a simple proof, although I think the following argument is far from optimal.