Let ( x n ) n ∈ N , ( y n ) n ∈ N {\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} },\,{\left(y_{n}\right)}_{n\in \mathbb {N} }} and ( z n ) n ∈ N {\displaystyle {}{\left(z_{n}\right)}_{n\in \mathbb {N} }} be three real sequences. Let x n ≤ y n ≤ z n {\displaystyle {}x_{n}\leq y_{n}\leq z_{n}} for all n ∈ N {\displaystyle {}n\in \mathbb {N} } and ( x n ) n ∈ N {\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }} and ( z n ) n ∈ N {\displaystyle {}{\left(z_{n}\right)}_{n\in \mathbb {N} }} be convergent to the same limit a {\displaystyle {}a} . Prove that also ( y n ) n ∈ N {\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }} converges to the same limit a {\displaystyle {}a} .
