Bounded, unbounded, convergent, divergent.

I am getting on ok with M208's book AA2 on sequences and I haven't come across any problems apart from one or two minor niggly points about some solutions to the exercises that I will eventually have to ask about. One of the nice things about this book is that I think I have finally got my head around the differences between the meanings of convergent/divergent, bounded/unbounded and tends to plus or minus infinity.

A sequence that tends to a limit L where L is a real number is convergent. If L is zero then this is a null sequence. If {a_n} is a sequence and |a_n|≤K for all n, for some positive number K, then the sequence is bounded. If the sequence is not bounded, then it is unbounded. If the sequence is convergent then it is bounded. If the sequence is unbounded then it is divergent. A sequence that tends to plus infinity is both unbounded and divergent. The same applies to a sequence that tends to minus infinity.

A point that should be made is that the converse of the statement "If the sequence is unbounded then it is divergent" is not true. The converse is "If the sequence is divergent then it is unbounded" and this is not true because bounded sequences such as {(-1)ⁿ} are divergent but not unbounded. Note also that some sequences such as {(-1)ⁿn} are unbounded but do not tend to either plus infinity or minus infinity.

I hope that makes it all clear for everyone. I can bet that in a month or two it will all become opaque for me again!