It is not necessary to close it, because BufferedWriter takes care of closing the writer it wraps.
To convince you, this is the source code of the close method of BufferedWriter:
public void close() throws IOException {
synchronized (lock) {
if (out == null) {
return;
}
try {
flushBuffer();
} finally {
out.close();
out = null;
cb = null;
}
}
}