Suppose f: [a,b] -> R and g: [a,b]-> R are both continuous. Let T={x: f(x)=g(x)}. Prove that T is closed.
I think the second way is a bit easier to do...
Since both f and g are continuous, you know that for a sequence such that , we have .
Now, think about this, if the sequence is from T, then we should have , right? Well, you just need to prove that is also in T. In other words, you need something like , which is true, but why?