Points in a unit square

Subsquares. Divide the unit square into $4$ equal subsquares.

Pigeonhole. $5$ points in $4$ subsquares: some subsquare contains at least $2$ points.

Distance. Two points in the same subsquare of side $1/2$ are at distance $\le\sqrt{(1/2)^2+(1/2)^2}=\frac{\sqrt2}{2}$.

Conclusion. Two points are within $\frac{\sqrt2}{2}$.