give a reference

A good reference is:

Dmitri Burago, Yuri Burago, and Sergei Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, Vol. 33, American Mathematical Society, 2001.

This is a standard metric-geometry reference; the available evidence lists the authors, publisher, series, volume, year, and ISBN information for the book . It is also recommended as a valuable source for concepts in metric geometry in notes on the Gromov–Hausdorff distance .

BibTeX

@book{burago2001course,
  title     = {A Course in Metric Geometry},
  author    = {Burago, Dmitri and Burago, Yuri and Ivanov, Sergei},
  series    = {Graduate Studies in Mathematics},
  volume    = {33},
  publisher = {American Mathematical Society},
  year      = {2001}
}

Sentence you can use

The Hausdorff distance is a metric on the collection of nonempty compact
subsets of a metric space, and hence satisfies the triangle inequality
\cite{burago2001course}.

If you want a direct proof reference

For a direct proof of the triangle inequality for the Hausdorff metric, the available evidence includes course notes that explicitly state “Now we prove the triangle inequality” for the Hausdorff metric on closed bounded subsets of (\mathbb{R}^n) . For a paper, however, I would cite the Burago--Burago--Ivanov book rather than homework notes.