[Mathematical Analysis for small readers] (proving that the series almost converges everywhere)

Set $ f \ In L (\ BBR) $ and test certificate: $ \ Bex \ VSM {n} f (N ^ 2x) \ EEx $ on $ \ BBR $, it almost converges to A Lebesgue function everywhere.

Proof:By $ f \ In L (\ BBR) $ Zhi $ | f | \ In L (\ BBR) $ (see [Cheng Qianyi, Zhang dianzhou, Wei Guoqiang, Hu shanwen, wang Shushi, basic of real-time function and functional analysis (Third edition), Beijing: Higher Education Press, 2010] Page 109 (VI )). since $ \ Bex \ VSM {n} \ int | f (N ^ 2x) | \ RD x = \ VSM {n} \ frac {1} {n} ^ 2 \ int | f (t) | \ rd t <\ infty, \ EEx $ we have (see [Cheng Qianyi, Zhang dianzhou, Wei Guoqiang, Hu shanwen, Wang Shushi, basis of Real Variable Function and functional analysis (Third edition), Beijing: higher Education Press, 2010] page 116 theorem 7) $ \ Bex \ int \ VSM {n} f (N ^ 2x) \ RD x = \ VSM {n} \ int F (N ^ 2x) \ RD x = \ VSM {n} \ frac {1} {n ^ 2} \ int F (t) \ rd t. \ EEx $ follow [Cheng Qianyi, Zhang dianzhou, Wei Guoqiang, Hu shanwen, Wang Shushi, basic of real-time function and functional analysis (Third edition), Beijing: Higher Education Press, 2010] Page 108 (ii) known $ \ Bex \ VSM {n} f (N ^ 2x) \ EEx $ on $ \ BBR $ is almost everywhere limited, while convergence.