I need a proof for this proposition:
> If a uniformity $U$ on $X$ has a
> countable fundamental system of
> entourages, then it can be defined by
> a pseudometric on $X$.

which is the proposition 2 on page 142 of [this book][1]. I can't see the proof page.

even a sketch of proof is welcome.
thanks.

  [1]: http://books.google.com/books?id=bQwhdmL6IjUC&pg=PA142&lpg=PA142&dq=if+a+uniformity+U+has+a+countable+fundamental+system+of+entourages,+then+there+is+a+pseudo+metric&source=bl&ots=eKHTzHWIG5&sig=H_H3KeyN-y_WcHhnxNMqShu5jf0&hl=en&sa=X&ei=aokuUcmFL9DitQbTuYDYAg&ved=0CD0Q6AEwAQ#v=onepage&q=if%2520a%2520uniformity%2520U%2520has%2520a%2520countable%2520fundamental%2520system%2520of%2520entourages%252C%2520then%2520there%2520is%2520a%2520pseudo%2520metric&f=false