Critic Ivor Smallbrain is sitting by the fire in his favourite pub, The Fox and Bounds. Also there are his friends Polly Gnomialle, Greta Picture, Gerry O'Laughing, Einstein, Hawking and celebrity mathematician Richard Thomas. Also joining them are great film directors Michael Loser and Ally Wooden.

All nine of them think that infinity is cool, and having seen the definition of what it means for a sequence \(\left(a_{n}\right)\) to have a limit \(a\), they are trying to define what it should mean to say that \(\lim a_{n}=\infty\). They decide that informally this should mean that you can make all the \(a_{n}\) 's as large as you like, provided you go far enough along the sequence. They then take it in turns to try to write down a proper rigorous definition. Here are their attempts:

Michael Loser writes: \(\forall a \in \mathbb{R}, a_{n} rightarrow \rightarrow a\).

Polly writes: \(\forall \varepsilon>0 \exists N \in \mathbb{N}\) such that \(n \geq N \Rightarrow\left|a_{n}-\infty\right|<\varepsilon\).

Greta writes: \(\forall R>0 \exists N \in \mathbb{N}\) such that \(n \geq N \Rightarrow a_{n}>R\).

Ally Wooden writes: \(\forall l \in \mathbb{R} \forall \varepsilon \in \mathbb{R} \exists N \in \mathbb{N}\) such that \(n \geq N \Rightarrow\left|a_{n}-l\right|>\) \(\varepsilon\).

Gerry writes: \(\forall a \in \mathbb{R} \exists \varepsilon>0\) such that \(\forall N \in \mathbb{N} \exists n \geq N\) such that \(\mid a_{n}-\) \(a \mid<\varepsilon\).

Einstein writes: \(\forall \varepsilon>0 \exists N \in \mathbb{N}\) such that \(\forall n \geq N, a_{n}>\frac{1}{\varepsilon}\).

Hawking writes: \(\forall n \in \mathbb{N}, a_{n+1}>a_{n}\).

Richard Thomas writes: \(\exists N \in \mathbb{N}\) such that \(\forall R>0, \forall n \geq N, a_{n}>R\)

Ivor writes: \(\forall R \in \mathbb{R} \exists n \in \mathbb{N}\) such that \(a_{n}>R\).

Who do you think is right and who do you think is wrong? (There may be more than one who is right!) For the wrong ones, illustrate why you think they are wrong with an example.