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My proof states, show that $H\subseteq H^{-1}$
Note: $H^{-1}$={$h^{-1}: h\in H$}
Here is my proof:
Let $h\in H$
We will show that $h\in H^{-1}$
Hence, $h^{-1}\in H$, because $h^{-1}\in H^{-1}$ of the subgroup of H
But $h^{-1}\in H$
Hence, $h\in H^{-1}$
therefore, $H\subseteq H^{-1}$
Is the right?

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    You should begin by saying "Let $h \in H."$ That is, just move the second line of the proof to the first line. – 2017-02-14
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    Is it correct now? – 2017-02-14

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Your wording isn't straightforward. Suppose $h \in H$. Because $H$ is a subgroup, it contains the inverse of any one of its elements. In particular, $k := h^{-1} \in H$. It then follows that $h = k^{-1} \in H^{-1}$. By the arbitrary nature of $h \in H$ we've proven the inclusion $H \subset H^{-1}$.

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    So my 1st line and 3rd line are not properly stated? To me it seems like I said the same thing you just stated – 2017-02-14
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    You did use the same technique. But in your proof, you write "Let $h \in H$. We will show that $h \in H^{-1}$. Hence...". The word "hence" doesn't really fit where you put it. It would be better in my opinion if you squeezed in another sentence. For instance, a slight modification would be "Let $h \in H$. We will show that $h \in H^{-1}$. To that end, we know $h^{-1} \in H$, because $H$ is a subgroup...." and then continue on from there. A few words make a big difference in the clarity of your proof. But ultimately, both of us did the same thing. I just worded mine a little differently. – 2017-02-14
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    Okay. So once I adjust, it will be proved? – 2017-02-14
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    There are a few things to adjust. You write "Hence $h^{-1} \in H$, because $h^{-1} \in H^{-1}$ of the subgroup $H$. Then $h = x^{-1}$ where $x \in H$. But $h ^{-1} \in H$..." This is a confusing passage. First you seem to be using the inclusion $H^{-1} \subset H$ to deduce $h^{-1} \in H$. Then you introduce an $x$ which is obviously equal to $h^{-1}$, but you never actually state what $x$ is equal to. But none of this matters, because the sentence "But $h^{-1} \in H..."$ seems to be deduced for reasons other than what is discussed in the sentences prior. Why not go straight to this sentence? – 2017-02-14
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    But yeah, judging from your original post, you know how to prove the desired result. Your wording is just off. – 2017-02-14
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    I adjusted the $x$ and use $h$. I was using my rough draft instead of my final draft – 2017-02-14
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    Gotcha. Writing $h = h^{-1}$ isn't quite right though. I think you meant $h=(h^{-1})^{-1}$. – 2017-02-14
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    that would be saying $h=h$ right? – 2017-02-14
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    Nothing needs to be said to deduce $h=h$; this is always true. I said $h = h^{-1}$ isn't quite right because nothing in your proof suggests that $h$ has to be its own inverse. – 2017-02-14
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    Okay. Thanks. Will adjust that too – 2017-02-14