From there you have to show that this mapping is well-defined, in the sense that it's independent of which sequence you choose that converges to a particular point. This isn't bad though. If then you know you can pick so large that they are both within ____ of and so by uniform continuity their images are with ____ of each other--ending in a for all and so they're equal. Note that this automatically gives you that (i.e. that extends ) since for each you can choose to be the constant sequence .
Showing its' continuous isn't bad then.
Does that help? If you're curious you can look on this blog post of mine, where I prove something (ever so) slightly more general.