asymptotic – Consistency of Jackknife and Bootstrap estimates for simple statistics

For both Jackknife and the simple Bootstrap, I'm supposed to test the consistency of the variance and the bias estimators. (Only for the simple case, that our statistics is a function of the sample mean).

I already have the tests for the Jackknife estimators, but it is difficult to find some for the Bootstrap (especially for the bias estimator).

My question is: We know that the Jackknife variance and the bias estimators are linear approximations of the respective Bootstrap estimates. Can not the results of the Jackknife estimators simply be transferred to the Bootstrap estimators, so that the Jackknife tests are not necessary?

(Also: I already proved that through the Jackknife bias correction, the first-order term in the bias disappears (again, the statistics are functions of the sample mean.) Can not we just transfer the result for the correction? of bias from Bootstrap?

Thanks for your help!