You're right! Here's my proof.

Suppose

is bounded and

takes Cauchy sequences to Cauchy sequences. I claim that

can be extended to a continuous function on

, which is compact; then

is uniformly continous on

and thus certainly on

also.

It suffices to show that if

are two Cauchy sequences with the same limit

, then

have the same limit which we may then call

. Consider the sequence

, which converges to

also; by assumption, it is mapped by

to a convergent sequence in

, which implies the subsequences

have the same limit!