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"within their bounds" is a correct and usable phrase in written English.
It can be used when referring to a situation or system in which some people or entities must operate or exist within a certain set of limitations or boundaries. For example, "The public school system must operate within their bounds of budget, personnel, and legal requirements."
Exact(9)
Nation states, then – and by definition – have commanded the loyalty of those who live within their bounds.
Abbey likely would have yawped with a kind of bitter pleasure when former Grand Canyon park director Stephen Martin told the Times that he thought Jarvis and the N.P.S. bent to pressure from Coca-Cola, which gives money to the parks and sells its Dasani water within their bounds.
The parks exist for the most part only as legal entities, however, as there has been little official control over any of them during the decades of social and political turmoil since their founding, and they have been left open to plant invasion and the establishment of subsistence agriculture within their bounds.
This century-old ruling gave the UK's local authorities control over the cinemas within their bounds, and though it was originally designed to ensure health and safety standards, it soon became the inadvertent legal basis for British film censorship.
Many of the most ardent supporters of a resolution to this conflict were New Jersey and Englewood Cliffs residents who cherish and take pride in the natural treasures within their bounds.
So, on the one hand, it might seem admirable that voters in a number of states have declared same-sex marriage legal within their bounds, even while other states are rejecting the same proposition.
Similar(51)
An alternate heuristic approach to maintain the variables within their allowable bounds involves the use of cascade controllers.
Physically intuitive discontinuous projection modifications are used to ensure all the on-line estimates within their known bounds.
Moreover, his responses to ancient and early modern critics of atomism are firmly within their strategic bounds, as they typically rely on reasoned argument without appeal to empirical force.
The design objective considered is to ensure that the top product, the bottom product, the reflux rate and the reboiler rate deviate within their prescribed bounds for all time and for all the disturbances.
To keep the input and output variables within their allowable bounds, the use of cascade controllers, and to track the optimal set of active input constraints, the use of split-range controllers is suggested in literature.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com