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To provide some insights on how to construct the region, we give the following simple example.

Is the usage of insights on correct in the above sentence ot it should be insights into ?

Please do not hesitate to suggest a better way to express the idea of the above sentence.

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It is definitely "to provide/give (...) insights on and not into. Instead of "to provide some insights on how to" I would prefer to say "to give you some insight on how to...". Note that I use insight as singular, especially since you are giving just one example!

I have one problem with the question though. I cannot imagine the context where one would construct a region. I have a feeling either to construct or region is not the ideal word to use here. If you could clarify this I can improve my answer.

EDIT So with "characterize the region" it would be:

To provide some insight on how to characterize the region, we give the following simple example.

To provide sounds rather formal to me, whereas the rest is more informal. A consistent formal way would be:

To provide some insight on how to characterize the region, consider the following example.

An informal example would be:

To get some insight on how to characterize the region, you may look at the following simple example.

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  • You can replace "construct the region" with "characterize the region".
    – din
    Jan 29 '17 at 17:20
  • @Michael Paul "Construct the region" would be a natural usage in geometry or topology, or some other branches of mathematics. It could also be used when defining borders, say in specifying an electoral district. Mar 18 '19 at 9:47
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According to this reference from the Canadian Government, to provide an insight about something is also possible. Insight into is also correct.

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  • well, in that case I am sorry
    – meso_2600
    Apr 24 '20 at 8:10

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