How to prove that, for a fixed $B \in(0,1)$, the sequence ${a_n}= (1/n).[nB] + n^2.B^n$ converges to $B$.

Question

For each $x\in \mathbb{R}$, let $[x]$ denote the greatest integer less than or equal to $x$.

Further, for a fixed $B \in(0,1)$, define, ${a_n}= (1/n).[nB] + n^2.B^n,$ for all $n\in N$.

Then, How I can show that the sequence ${a_n}$ converge to $B$.

Plz help !!

Answer 1

Hint:

$$ B - \frac{1}{n} [ nB ] = \frac{ nB - [nB] }{n} $$