Is the sentence "It is not enough that I should succeed – others should fail" correct? I mainly wonder about the dash. Should it be a comma instead?
Does anything else need to be changed?
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Well, it should have a period at the end. (Or if you'd rather, an exclamation point!)
As for the hyphen: use whatever you please. You can find someone here to defend whatever symbol and spacing you use, and to quote a reputable style guide to back them up.
So I could give advice on this, but it seems to raise tempers here. Let's just say that if you're confused, you're in good company.
If you are able, on your device, to create an em-dash, you could use an em-dash.
If you are able to create an en-dash, you could use an en-dash.
If you are able to create hair spaces, you could use hair spaces.
If you have a style guide you prefer to follow, you could follow that style guide.
If you are able to ignore all the advice you read here about hyphens and dashes, you might be better off.
IAMOS (It's a matter of style). In other words, POB.