I would like an easy example of an infinite simple group along with an embedding into a finitely presented group. I know that there are infinite simple finitely presented group such as Thompson’s group $V$ and I know that any group with solvable word problem embeds into a simple subgroup of a finitely presented group, but I am looking for an example of a more elementary nature. 



Easy example of an infinite simple group with an embedding into a finitely presented group
I think D. L. Johnson's article *[Embedding some recursively presented groups](https://www.cambridge.org/core/books/groups-st-andrews-1997-in-bath/embedding-some-recursively-presented-groups/A5CF8D51C4641ECB739C7A8CBDC98531)* should answer your question. The abstract is:



> 
> We seek to illustrate the Higman Embedding Theorem by finding actual embeddings
>  of various popular recursively presented groups in finitely presented ones,
>  and are successful in at least one case.
> 
> 
> 


In particular, Section 3 contains a short and elementary proof of the fact that the group $S\_\infty$ of bijections of $\mathbb{N}$ with finite supports (which contains the infinite simple group $A\_\infty$ of alternating bijections) embeds into a finitely presented group. A presentation for such a group, simplified in Section 4, is
$$\langle a,b,x \mid a^2=1, (xaxa^{-1})^3=1, [x,a^2xa^{-2}]=1, x=[a,b], axa^{-1}=bxb^{-1} \rangle.$$
Actually, the group defined by the presentation coincides with Houghton's group $H\_3$ already mentioned by Yves in the comments.


