algebra precalculus - Proving that a series is bounded from above through induction.

I am required to prove that the following series

$$a_1=0, a_{n+1}=(a_n+1)/3, n \in N$$ is bounded from above and is monotonously increasing through induction and calculate its limit. Proving that it's monotonously increasing was simple enough, but I don't quite understand how I can prove that it's bounded from above through induction, although I can see that it is bounded. It's a fairly new topic to me. I would appreciate any help on this.

Answer

See that for any $a_n<\frac12$, we have

$$a_{n+1}=\frac{a_n+1}3<\frac{\frac12+1}3=\frac12$$

Thus, it is proven that since $a_0<\frac12$, then $a_1<\frac12$, etc. with induction.

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We choose $\frac12$ since, when solving $a_{n+1}=a_n$, we result with $a_n=\frac12$, the limit of our sequence.