Answer: You're on the right track. Suckers will continue to sprout as long as the roots remain alive. If you keep cutting the suckers down (before they produce leaves), the roots will eventually run out of energy and will suckers will cease to sprout. You can help things along by applying full-strength glyphosate (such as RoundUp) to the ends of the sprouts immediately after you cut them. Glyphosate will be translocated to the roots and will kill at least a small portion. If you keep at it, you'll eventually kill off all the roots. Use a paintbrush to apply the glyphosate to the cut ends and be careful - it is a contact killer so anything it touches will die, including desired grass, groundcover, etc.
Best wishes with your new landscape.
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