In working on coursework I had a thought about recursive sequences. For monotone. bounded, recursive sequences $(x_n)$ , $\lim_{n\to\infty}(x_n)$ seems to equal the point where the differences in terms is zero (i.e. $x_n-x_{n+1}=0$). On brief consideration it seems to stand to reason - the term difference is essentially the rate of change of the sequence, a monotone sequence must be changing in the same direction for all terms. Thus, the sequence should tend toward the point where the next term would be larger if the sequence got there, but never quite actually get there. Hence, the limit.
I would like to ask: is this a known result? If so, does the theorem have a name? If not, where can one poke holes in the claim?