How to show completeness of C(K)?

[minibear]

The thing is: if you have (K, d) to be a compact metric space, then how to show (C(K) , d_infinity) is complete ?( d_infinity is uniform metric)

I know maybe I should use Cauchy or triangle inequality, but I am just not clear how to construct the proof.

Anyone who has any idea about that, please help. THX!

 

[HallsofIvy]

To show that any metric space is complete, you must show that all Cauchy sequences converge, the definition of "complete".

Now, since C(k) is the set of continuous functions on K, with "sup" metric, each member of C corresponds to a continuous function and a Cauchy sequence on C(k) is a sequence of continuous functions on K, say {f_n}. I would look at the sequence {f_n(x)} for each x in K. Try to show that is a Cauchy sequence in K.