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How do I prove this statement?

Let $G$ be a Topological group and let $H$ be a subgroup of $G$, if both $H$ and $G/H$ are locally compact then $G$ is locally compact. (we will endow the set $G/H$ with the quotient topology)

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    5.25 in Hewitt-Ross' Abstract Harmonic Analysis I. I don't know of any "instant" proof...2012-04-18
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    Is my answer good enough, or do you need clarifications?2012-05-22
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    quick question. how do we know that $G/H$ is locally compact for sure?2014-02-24

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