If a sequence is unbounded will it necessarily diverge?

Are you sure the question isn't that "not every bounded series converges"? For example

\+ 1 - 1 + 1 - 1 + 1 ...
The problem is the definition of 'diverge'
Some say that 'diverge' means not having a *finite* limite, so this excludes infinities
Some say that diverge means 'having infinity as its limit'
It depends on the definition of diverge. In some classes it means go to infinity, in some it just means "not converge". Seems like they're thinking the first one. So pick a sequence that's unbounded, but not all the points go that way.
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**if a sequence is unbounded it will diverge**, this is **true**.

the contra positive of this statement is **if a sequence converges it is bounded,** is equivalent and thus also **true**

the inverse  of this statement, that **if a sequence is bounded it will converge**  is **false** (counterexample: 1,-1,1,-1,1,-1...)

and the converse of this statement that **if a sequence is diverges it is unbounded** is **false**
(counterexample 1,-1,1,-1,1,-1...)
yes, when talking about sequences in the real numbers, every unbounded sequence diverges.