The spatial resolution of the focused field of a classical time-reversal mirror has a wavelength-order lambda diffraction limit. Previously reported results for subwavelength focus require either the full knowledge of the original source or the evanescent waves in the near field. Here it is shown that subwavelength focusing can be achieved without a priori knowledge of the original probe source. If the field is recorded at a few wavelengths away from the probe source, where the amplitude of the near field is too low for subwavelength focusing, it is shown that the low amplitude near field can be amplified and the spatial resolution improved, using the near-field time reversal (NTR) procedure introduced here. The NTR is performed from the phase of the spatial spectrum of the field recorded on an array around the original probe source using an analytical continuation for the amplitude of the spatial spectrum. Following theory, lambda/20 resolution is experimentally demonstrated with audible acoustic wavefields in the air.