Let X be a Hausdorff space and Y be a subset of X. Then, Y with the subspace topology is a Hausdorff space. 
Question: Let X be a Hausdorff space and Y be a subset of X. Then, Y with the subspace topology is a Hausdorff space. 
This is what I did, can someone verify this and let me know if I am correct or wrong? Also, kindly let me know if my proof need some changes or modifications due to bad notations. 
Any help will be greatly appreciated.

Everything that needs to be there is there, so it's a valid proof.  My only comments are about the style.


*

*The line where you recall the definition of the subspace topology on $Y$ is out of place.  You've already used this definition once; either you should state it at the top before you use it the first time, or leave it out altogether.

*“Hence a set containing $x$ in $Y$ is $U' = U \cap Y$, which is open in $Y$” might read better if it were written as “Hence $U' = U \cap Y$ is open in $Y$ and contains $x$.”  You're defining $U'$ in this sentence, so I feel like that definition should come first, and its properties later.
Otherwise good job!
