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We present an analytical formulation that shows the negative refraction of propagating waves and the growth of evanescent waves within a negative refractive index (NRI) lens made of a periodically L, C loaded transmission-line (TL) network, referred to as the dual-TL structure. A transformation known as the "array scanning method" is then employed to analytically demonstrate the sub-diffraction imaging capability of a dual-TL lens. In essence, the two-dimensional (2-D) periodic Green's functions corresponding to the voltages and currents excited by a vertical elementary current source are derived. The developed theory is utilized to plot the 2-D voltage magnitude distribution for the case of focusing an elementary current source. The analysis reveals that a resolution limit is imposed by the periodicity of the NRI medium used. Moreover, the periodicity of the NRI medium bounds the amplitude of the growing evanescent waves in a realizable NRI lens and prevents them from growing to unphysically large values.