1 1. Remember: you are writing for an expert. Cross out all that is trivial or routine. 2 3 2. Avoid repetition: do not repeat the assumptions of a theorem at the beginning of its proof, or a complicated conclusion at the end of the proof. Do not repeat the assumptionos of a previous theorem in the statement of a next one (instand, write e.g."Under the hypotheses of Theorem 1 with f replaced by g,....."). Do not repeat the same formula -- use a label instead. 4 5 3. Check all formulas: is each of them necessary?
We denote by $mathbb{R}$ the set of all real numbers.
We have the following lemma.
The following lemma will be useful.
...... the following inequality is satisfied：
We denote by $mathbb{R}$ the set of all real numbers.
We have the following lemma.
The following lemma will be useful.
...... the following inequality is satisfied：
1 Let $varepsilon$ be an arbitrary but fixed positive number $Rrightarrow$ Fix $varepsilon>0$ 2 3 4 5 Let us fix arbitrarily $xin X$ $Rrightarrow$ Fix $xin X$ 6 7 8 9 Let us first observe that $Rrightarrow$ First observe that 10 11 12 13 We will first compute $Rrightarrow$ We first compute 14 15 16 17 Hence we have $x=1$ $Rrightarrow$ Hence $x=1$ 18 19 20 21 Hence it follows that $x=1$ $Rrightarrow$ Hence $x=1$ 22 23 24 25 Taking into account (4) $Rrightarrow$ By (4) 26 27 28 29 By virtue of (4) $Rrightarrow$ By (4) 30 31 32 33 By relation (4) $Rrightarrow$ By (4) 34 35 36 37 In the interval $[0,1]$ $Rrightarrow$ in $[0,1]$ 38 39 40 41 There exists a function $fin C(X)$ $Rrightarrow$ There exists $fin C(X)$ 42 43 44 45 For every point $pin M$ $Rrightarrow$ For every $pin M$ 46 47 48 49 It is defined by the formula $F(x)=......$ $Rrightarrow$ It is defined by $F(x)=......$ 50 51 52 53 Theorem 2 and Theorem 5 $Rrightarrow$ Theorems 2 and 5 54 55 56 57 This follows from (4),(5),(6) and (7) $Rrightarrow$ This follows from (4)-(7) 58 59 60 61 For details see [3],[4] and [5] $Rrightarrow$ For details see [3]-[5] 62 63 64 65 The derivative with respect to $t$ $Rrightarrow$ The $t-$ derivative 66 67 68 69 A function of class $C^2$ $Rrightarrow$ A $C^2$ function 70 71 72 73 For arbitrary $x$ $Rrightarrow$ For all $x$ (For every $x$) 74 75 76 77 In the case $n=5$ $Rrightarrow$ For $n=5$ 78 79 80 81 This leads to a constradiction with the maximality of $f$ $Rrightarrow$ .....,contrary to the maximality of $f$ 82 83 84 85 Applying Lemma 1 we conclude that $Rrightarrow$ Lemma 1 shows that ......, which completes the proof $Rrightarrow$ .......$Box$
Let $varepsilon$ be an arbitrary but fixed positive number $Rrightarrow$ Fix $varepsilon>0$
Let us fix arbitrarily $xin X$ $Rrightarrow$ Fix $xin X$
Let us first observe that $Rrightarrow$ First observe that
We will first compute $Rrightarrow$ We first compute
Hence we have $x=1$ $Rrightarrow$ Hence $x=1$
Hence it follows that $x=1$ $Rrightarrow$ Hence $x=1$
Taking into account (4) $Rrightarrow$ By (4)
By virtue of (4) $Rrightarrow$ By (4)
By relation (4) $Rrightarrow$ By (4)
In the interval $[0,1]$ $Rrightarrow$ in $[0,1]$
There exists a function $fin C(X)$ $Rrightarrow$ There exists $fin C(X)$
For every point $pin M$ $Rrightarrow$ For every $pin M$
It is defined by the formula $F(x)=......$ $Rrightarrow$ It is defined by $F(x)=......$
Theorem 2 and Theorem 5 $Rrightarrow$ Theorems 2 and 5
This follows from (4),(5),(6) and (7) $Rrightarrow$ This follows from (4)-(7)
For details see [3],[4] and [5] $Rrightarrow$ For details see [3]-[5]
The derivative with respect to $t$ $Rrightarrow$ The $t-$ derivative
A function of class $C^2$ $Rrightarrow$ A $C^2$ function
For arbitrary $x$ $Rrightarrow$ For all $x$ (For every $x$)
In the case $n=5$ $Rrightarrow$ For $n=5$
This leads to a constradiction with the maximality of $f$ $Rrightarrow$ .....,contrary to the maximality of $f$
Applying Lemma 1 we conclude that $Rrightarrow$ Lemma 1 shows that ......, which completes the proof $Rrightarrow$ .......$Box$