Write down an example of a sequence ${a_n}$ such that ${a_n}$ is divergent, $a_n < 5$ for all $n$, and $a_n < a_{n+1}$ for all $n$

Question

Write down an example of a sequence ${a_n}$ such that:

${a_n}$ is divergent

$a_n < 5$ for all $n$, and

$a_n < a_{n+1}$ for all $n$,

or explain why no such divergent sequence exists.

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What I did thus far:

We know that the sequence is increasing since we are given that $a_n < a_{n+1}$ for all $n$, thus we can also say that it is monotonic.

We also know that the sequence is bounded from above since we are given that $a_n < 5$ for all $n$.

By the Monotonic Sequence Theorem, we know that every bounded and monotonic sequence is convergent.

Thus, no such divergent sequence exists.