A half-life is the time taken for the radioactivity of a material to fall to half its original value. A material can undergo infinite half-lives because each time it falls to half the next half-life falls to half of that half:

No half-lives have elapsed when radioactivity is at the original amount; 1/1.

1 half-life is when radioactivity is at 1/2

2 half-lives is when radioactivity is at 1/4.

3 half-lives is when radioactivity is at 1/8.

4 half-lives is when radioactivity is at 1/16.

And so on.

1/8 amount of the element will reamain.

After 1 half-life there is 1/2 left

After the 2nd half-life there is 1/2 of 1/2 left which is 1/4

After the 3rd half-life there is 1/2 of 1/4 left which is 1/8

Half of the original sample of a radio isotope remains after a half-life period. After two half-life periods, one-fourth of the radio isotope remains.

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No, it is not. If a radionuclide has a half-life of 1 month, half is gone after 1 month. Half of the half that is left will be gone after 2 months, and that will leave 1/4th of the original amount left after the second month.