A uniformly continuous on the unit sphere of $c_0$ is bounded Let $S_{c_0}= \{ x\in c_0 \;: \Vert x \Vert =\;1\; \}$, where $c_0$ is the space of all sequences converging to zero and 
$$f: S_{c_0} \rightarrow \mathbb R$$
is a uniformly continuous function. Prove that $f$ is  bounded. 
I thought that if I could show $S_{c_0}$ is a compact set then by using the continuity of $f$, I would result in the requested point, but I feel like I'm missing something. 
Could anyone please help or give some hints? Thanks in advance!

The unit sphere of $c_0$ is certainly not compact as the set $\{e_n\colon n\in \mathbb{N}\}\subset S_{c_0}$ is closed and discrete. (The unit sphere of a normed space is compact if and only if the space is finite-dimensional.)
However, you may adapt this proof to see that that the unit ball as well as the unit spehre of a normed space has the property that each uniformly continuous function on it is bounded.
