Simple proof question (limits)

[htaMandPhysics]

I was reviewing some of the material in my calculus textbook when I realized that I really struggle with writing proofs. Therefore, I decided to attempt one of the easier proof related exercises found in the book. I would ask my teacher for help, but I am off for summer break.

Homework Statement

Prove that if the limit x->c of f(x) exists and limit x->c of [f(x)+g(x)] does not exist, then limit x->c of g(x) does not exist

Homework Equations

The Attempt at a Solution

Based on the properties of limits, if limit x→c of f(x)=L and limit x→c of g(x)=K, then the limit x→c of [f(x)+g(x)]=L+K. So let us assume that the limit x→c of g(x) does exist and is equal to K, and the limit x→c f(x) is equal to L. If that is so, then limit x→c of [f(x)+g(x)] must exist and is equal to L+K. However this creates a contradiction, as it was stated that limit x→c of [f(x)+g(x)] does not exist and therefore ≠ L+K. Therefore lim x→c of g(x) can't exist, else a contradiction is created.

So I tried to do a "proof by contradiction"... was I successful? If not, where exactly did I go wrong? Thanks

 

[HallsofIvy]

Yes, that is exactly right.