This means, . It must be the case that because the sequence of ratios is a sequence of non-negative terms. So the limit is non-negative. I leave the case to you to prove. I will assume , i.e. . The important step is to notice that there most exists so that . This means for we have by convergence, for . Thus, for .
This means,
.
In general,
for .
This tells us that the sequence is bounded by for all .
Thus,
.
Now, .
Because and it is a geometric sequence.
Q.E.D.