Solved – Consistency of sample quantile

I'm new to statistics and recently I learnt about non-parametric estimation. On the estimation of empirical quantiles, many notes talk about the asymptotic behavior of sample quantiles. The proof is based on the consistency of the empirical quantile estimator, then applying Slutsky's theorem in combination with the delta method (assuming the cdf is $C^{1}$ at the considered point).

However, it seems to give a very old and trivial result stating that the empirical quantile converges in probability to the true value (with $C^1$ cdf hypothesis). I've asked my professor and he just give the suggestion using Hoeffding's inequality (he doesn't want to spend his time explaining such an easy problem). So, could someone please provide me the proof of this, or a reference covering this issue (e.g. a note on the internet, or link to a pdf file.)

The proof using Hoeffding's inequality can be found at page 8 of these notes.

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