Let ( x n ) n ∈ N {\displaystyle {}(x_{n})_{n\in \mathbb {N} }} and ( y n ) n ∈ N {\displaystyle {}(y_{n})_{n\in \mathbb {N} }} be sequences of real numbers and let the sequence ( z n ) n ∈ N {\displaystyle {}(z_{n})_{n\in \mathbb {N} }} be defined as z 2 n − 1 := x n {\displaystyle {}z_{2n-1}:=x_{n}} and z 2 n := y n {\displaystyle {}z_{2n}:=y_{n}} . Prove that ( z n ) n ∈ N {\displaystyle {}(z_{n})_{n\in \mathbb {N} }} converges if and only if ( x n ) n ∈ N {\displaystyle {}(x_{n})_{n\in \mathbb {N} }} and ( y n ) n ∈ N {\displaystyle {}(y_{n})_{n\in \mathbb {N} }} converge to the same limit.
