The simplest way to prove that any left-invariant vector field on a Lie group is complete

It's all in the question: I look for the most intuitive proof that the integral curves of any left-invaraint vector field on a Lie group can be extended for all values of "time". I realize that the argument is always based on the existence of group multiplication; what I look for is the most straightforward proof available. Thanks in advance!

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I do think this statement is true if $G$ is not connected. If $G$ is connected and compact, then you can prove this via Hopf-Rinow theorem using the exponential map. See Terence Tao's article on this in his blog.