A universal fluctuation diamagnetic susceptibility in cuprate superconductor Bi$_{\mathrm{\mathbf{2}}}$\textbf{Sr}$_{\mathrm{\mathbf{2-x}}}$\textbf{La}$_{\mathrm{\mathbf{x}}}$\textbf{CuO}$_{\mathrm{\mathbf{6+\delta }}}$

As we know, superconducting fluctuation appears at temperatures much higher than the superconducting transition temperature $T_{\mathrm{c}}$ in cuprate superconductors. It is not clear yet that how these fluctuations evolve into a superconducting condensation. In this work, we use torque magnetometer, a sensitive tool to detect magnetic signal, to study these fluctuations in Bi$_{\mathrm{2}}$Sr$_{\mathrm{2-x}}$La$_{\mathrm{x}}$CuO$_{\mathrm{6+\delta \thinspace }}$and found a universal condition under which the superconducting transition will occur, no matter the superconductor is underdoped, optimal doped or overdoped. This condition is coincident with a Bose-Einstein condensation (BEC), which reveal the nature of superconductivity in cuprate is of BEC type. The preformed cooper pairs above $T_{\mathrm{c}}$ can be viewed as bosons and follows a correlation function, which determined the strength of fluctuations.