Find lim sup and lim inf of $((-1)^n+1)+1/(2^n)$

Question

Here is what I have, but I do not know if my understanding of lim sup and lim inf are correct:

lim sup= 2 because the smallest sup of the sequence would be $(1+1)+0=2$

lim inf=0 because the largest inf of the sequence would be $(-1+1)+0=0$

Is this the idea or am I off base?

Answer 1

The answer is correct, but I don't know what is that you have in mind when you talk about the “smallest sup” and about the “largest inf” of the sequence. The superior limit of the sequence is $2$ because $2$ is the biggest real number which is the limit of a subsequence of your sequence and the inferior limit of the sequence is $0$ because $0$ is the smallest real number which is the limit of a subsequence of your sequence