Let $(a_{n})_{n=1}^{\infty}$ be a bounded sequence, and suppose that the sequence $(a_{n}+ 2a_{n+1})_{n=1}^{\infty}$ converges to $3L \in R$. How can I prove that $(a_{n})_{n=1}^{\infty}$ converges to $L$ ?
How can I go about proving this? I tried using the definition of a limit for the 2 sequences, and tried iterating the inequalities, but I did not seem to be getting anywhere. Could someone provide a proof for this problem, I am really trying my best to understand a proof for this problem.
Also, is there any way to prove the above without, assuming that $(a_{n})_{n=1}^{\infty}$ is bounded.
Really would appreciate your help on this problem. Thank You!