Here are some other ways to say it:
When you believe in a cause, you'll do anything to see it accomplished.
When you believe in a cause, you'll pursue it to the end. (pursue is like chase but works for abstract goals not just people or cars)
When you believe in something, you'll give it your all.
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When you believe in a cause, you have to give it your all.
When you believe in something, you have to see it accomplished.
When you believe in something, you can't stop until it's done.