Corrected! Thank you!

I'm really glad you're enjoying the website! You're right to be skeptical of the statement and the proof though!

The "Squeeze Theorem" has a few variations. The general version for sequences is that if $(a_n)$, $(b_n)$, and $(c_n)$ are sequences of real numbers and there exists some number $M$ such that $a_n \leq b_n \leq c_n$, for all $n \geq M$ and if $(a_n)$ and $(c_n)$ converge to $L$ then $(b_n)$ must also converge to $L$.

Some texts and teachers give the simpler (less general) version where it is assumed that $a_n \leq b_n \leq c_n$ for every $n$. (The same conclusion holds).

I stated the more general version but proved the simpler version. Thank you for spotting this little error! I will correct it immediately.