$a_n>0$, and the following attitude is true. Prove that the sequence is convergent.
So, what I tried here is to simply write down the elements of the sequence. If we have a "big enough" $n$, then $\frac12^n$ will converge to $0$, so what we have left is $a(n) < a(n-1)$ (the $n$-th element is smaller than the $n-1$-th element), which means that this sequence monotonically decreasing, after a big enough $n$.
Since this is a monotonically decreasing sequence, if we can find a number as a limit, then it must be convergent, but I can't find any.
Am I thinking wrong or what should be the correct way?