First let me give you the definitions for "Limit of a sequence" and "Boundedness".
Limit of a sequence.
A sequence is said to converge to if such that whenever
Boundedness of a sequence.
A sequence is said to be bounded if such that,
Therefore boundedness and the limit tending to infinity is not the same. Anyhow it could be shown that every sequence which is convergent (has a limit) is bounded. I am also doing Real analysis in our university. Therefore if you have anything to discuss regarding this matter please don't hesitate to do so.
Hope this helps.