The runner is not out of bounds - the definition makes that pretty clear. However, NF 9-1 reads: "An offensive player shall not push, pull or lift the runner to assist his forward progress."
I will admit that this is only my interpretation, but I would say that pushing the runner to keep him inbounds is an assist to his forward progress, and should draw a flag for helping the runner. Whether I would have the presence of mind to throw it if this actually happened, that I'm not so sure of.
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