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I have this formula ##\left[2\sqrt{Q\left(\sqrt{2\eta}\right)}\right]^N## where Q is the Q Gaussian function which can be upper bounded by the Chernoff bound ##Q\left(\sqrt{2\eta}\right)\leq exp\left(-\eta\right)##, and thus the original formula can be upper bounded as ##2^Nexp\left(-\frac{N\,\eta}{2}\right)##. Right? Now I need this upper bound to be less than a certain value, say ##\varepsilon\leq 0.5##, i.e. ##2^Nexp\left(-\frac{N\,\eta}{2}\right)\leq \varepsilon##. It seems straightforward to solve this inequality to find N for a given ##\eta##. However, when I did, and did the verification, the term ##2^Nexp\left(-\frac{N\,\eta}{2}\right)## isn't less than ##\varepsilon##. Do I need to be careful about something here that I might have missed?

Thanks