## Example 3.3.5(a):

Since*| sin(x) | < 4*, the sequence is clearly bounded above and below (the sequence is also, of course, bounded by 1).

Therefore, using the Bolzano-Weierstrass theorem, there exists a convergent subsequence.

However, it is nearly impossible to actually list this subsequence. The Bolzano-Weierstrass theorem does guaranty the existence of that subsequence, but says nothing about how to obtain it.

The original sequence *{ sin(j) }*, incidentally, does not
converge. The proof of this is not so easy, but if we assume that the
second part of this example has been proved, it would be easy. Remember
that the second part of this example states that given *any*
number *L* with *|L| < 1* there exists a
subsequence of
that converges to *L*. If that was true the original sequence
can not converge, because otherwise all its subsequences would have to
converge to the same limit.

Of course this proof is only valid if this - more complicated - statement can be proved.