Hi I am having a bit of trouble expressing the answer to a problem. If it is any constellation this problem goes alongside convergence proofs.

$\displaystyle Let \{a_n \}^\infty_1 , \{b_n\}^\infty_1 $ be sequences such that $\displaystyle lima_n = a$ and $\displaystyle b_n = a_n $ if n is even

or $\displaystyle b_n = a$ if n is odd. Show that $\displaystyle limb_n = a $

That being said if n is even $\displaystyle b_{2n} = a_2 + a_4 + ... + a_{2n} $ . As $\displaystyle a_{2n} $ is a subsequense of $\displaystyle a_n $ one can conclude that $\displaystyle a_{2n} $ converges to $\displaystyle a $. Hence $\displaystyle b_n $ will converge to a if n is even.

As $\displaystyle b_{2n+1} = a $ is a is a constant. $\displaystyle b_{2n+1} $ will also converge to $\displaystyle a $.

As $\displaystyle b_n $ converges to a in both cases, then $\displaystyle limb_n = a $ as stated.

I think this is the idea behind the question, if I overlooked anything please tell me. Thank you.