Following construction is to be made in the given figure.

Extend QS to meet PR at T.

Now, in ∆ PQT,

PQ+PT>QT (sum of two sides is greater than the third side in a triangle)

PQ+PT>SQ+ST (i)

Now, In ∆ STR,

ST+TR>SR (ii)(sum of two sides is greater than the third side in a triangle)

Now, adding (i) and (ii),

PQ+PT+ST+TR>SQ+ST+SR

PQ+PT+TR>SQ+SR

PQ+PR>SQ+SR

SQ+SR<PQ+PR

Hence, proved