It's evident that there is a bijection from (0,1) to (0,1) U NAT from the inclusion map. But to show an injection the other way, I tried mapping all n in NAT to 1/(n+1) but then there was no space for mapping elements of the form 1/n from (0,1) to (0,1).
Can someone please guide me to come up with an injection?
Here's one of many ways to Proceed. You can inject (0,1) into an open interval contained within (0,1) -- think of squeezing it. Then there will be plenty of room left in the open interval to inject the natural numbers. In fact, you should be able to inject the naturals into any open interval. (First demonstrate that. Then use it to verify "There will be ... the natural numbers."
If you can picture the above, it will become "obvious" to you.