Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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Terence Tao, Analysis 1. Exercise 5.3.2. Real Numbers and Cauchy Sequences.

Let $ x = \lim_{n\rightarrow\infty}a_n, y = \lim_{n\rightarrow\infty}b_n$, and $ x' = \lim_{n\rightarrow\infty}a'_n$ be real numbers. Then $xy$ is also a real number. Furthermore, is $x=x'$, then $xy ...
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Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $b>a$, $a>0$, ...
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show that: $f$ is injective $\iff$ there exists a $g: Y\rightarrow X$ such that $g \circ f = idX$

** proof under construction - will post when done and more or less confident it's true. ** also please easy with the downgrades.. i don't understand why i'm getting them. what is meant by show ...