Extremely large antenna arrays envisioned for 6G incurs near-field effect, where steering vector depends on angles and range simultaneously. Polar-domain near-field codebooks can focus energy accurately but incur extra two-dimensional sweeping overhead; compressed-sensing (CS) approaches with Gaussian-masked DFT sensing offer a lower-overhead alternative. This letter revisits near-field beam training using conventional DFT codebooks. Unlike far-field responses that concentrate energy on a few isolated DFT beams, near-field responses produce contiguous, plateau-like energy segments with sharp transitions in the DFT beamspace. Pure LASSO denoising, therefore, tends to over-shrink magnitudes and fragment plateaus. We propose a beam-pattern-preserving beam training scheme for multiple-path scenarios that combines LASSO with a lightweight denoising pipeline: LASSO to suppress small-amplitude noise, followed by total variation (TV) to maintain plateau levels and edge sharpness. The two proximal steps require no near-field codebook design. Simulations with Gaussian pilots show consistent NMSE and cosine-similarity gains over least squares and LASSO at the same pilot budget.