Lemma: For any integer n > 0, we have

Lemma: For we have

Lemma: For all real numbers x ≥ 1, we have x # < 4 < sup > x </ sup >.

: Fekete's Subadditive Lemma: For every subadditive sequence, the limit exists and is equal to.

Lemma 1: For all codewords, in a Walsh – Hadamard code,,, where represent the bits in in positions and respectively, and represents the bit at position.

The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.

Conditions for Lemma 2 shall hold for all factor, any violation will result in satisfying the condition of Lemma 1 whereby the result has been proven.

: Lemma: The sum of any non-empty set of distinct, non-consecutive Fibonacci numbers whose largest member is is strictly less than the next largest Fibonacci number.

* Franz Miklosich ( Lemma by Katja Sturm-Schnabl, p. 186-193 ) in: Marija Mitrović, Die Geschichte der slowenischen Literatur von den Anfängen bis zur Gegenwart.

Proof: Using the same sum for R ( p, n ) as in Lemma 2, we see that since p < sup > 2 </ sup > > 2n, in fact only the first term is nonzero ; this term is exactly the right-hand side of the above equation.

Further, we see by Lemma 2 that the multiplicity of F can only change at a tangent point, and at such a point can only change by an even integer.

As shown by the Lemma above, we only need to prove our theorem for formulas φ in R of degree 1. φ cannot be of degree 0, since formulas in R have no free variables and don't use constant symbols.

The Yoneda Lemma states that is fully faithful and we also get the left exactness very easily because is already left exact.

We have proved that φ is either satisfiable or refutable, and this concludes the proof of the Lemma.

Her performances of Xenakis's Herma, and Brian Ferneyhough's Lemma Icon Epigram have received critical acclaim.

It follows from an application of Gauss's Lemma that if A is the norm of then the distance, induced by the metric, between two close enough points on the curve γ, say γ ( t < sub > 1 </ sub >) and γ ( t < sub > 2 </ sub >), is given by

Lemma: A number n is k-hyperperfect ( including k = 1 ) if and only if the k-hyperdeficiency of n, δ < sub > k </ sub >( n )

Lemma: Let A ∈ C < sup > n × n </ sup > be a complex-valued matrix, ρ ( A ) its spectral radius and ||·|| a consistent matrix norm ; then, for each k ∈ N:

Lemma ( 15. 8, Rudin ) for | z | ≤ 1, n ∈ N < sub > o </ sub >

We may therefore apply Zorn's Lemma to conclude that A has a maximal element, say ( M, R ).

Lemma: A number n is k-hyperperfect ( including k =

With Ajtai, Chvátal, and M. M. Newborn, Szemerédi proved the famous Crossing Lemma, that a graph with n vertices and m edges, where has at least crossings.

Lemma 1.

Lemma 2.

Lemma 3.

otherwise by Lemma 1 the component would extend beyond these points.

Lemma 4.

* Lemma A Banach space X is reflexive if and only if the natural pairing on X × X ′ is perfect.

Lemma.

However, this idea can be understood as a basis for the following proof of the Lemma.

Lemma 1.

Lemma 2.

Pavel Samuilovich Urysohn, Pavel Uryson () ( February 3, 1898, Odessa – August 17, 1924, Batz-sur-Mer ) was a Jewish mathematician who is best known for his contributions in the theory of dimension, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology.

( The same applies to Zorn's Lemma.

The well ordering theorem follows easily from Zorn's Lemma.

In this way, Yoneda's Lemma provides a complete classification of all natural transformations from the functor Hom ( A ,-) to an arbitrary functor F: C → Set.

He also gives starting at Lemma 4 and Proposition 40 ) the theory of the motions of comets ( for which much data came from John Flamsteed and from Edmond Halley ), and accounts for the tides, attempting quantitative estimates of the contributions of the Sun and Moon to the tidal motions ; and offers the first theory of the precession of the equinoxes.

Ngô Bảo Châu proved an auxiliary but difficult statement, the so-called " Fundamental Lemma ", originally conjectured by Langlands.

For the answer cannot be derived from any socially cohesive element in the disrupting community.

For the truth formerly experienced by the community no longer has existential status in the community, nor does any answer elaborated by philosophers or theoriticians.

William Wimsatt and Cleanth Brooks, it seems to me, have a penetrating insight into the way in which this control is effected: `` For if we say poetry is to talk of beauty and love ( and yet not aim at exciting erotic emotion or even an emotion of Platonic esteem ) and if it is to talk of anger and murder ( and yet not aim at arousing anger and indignation ) -- then it may be that the poetic way of dealing with these emotions will not be any kind of intensification, compounding, or magnification, or any direct assault upon the affections at all.

For if Serenissimus made the sign of the Cross with his right hand, and meant it, with his left he beckoned lewdly to any lady who happened to catch his eye.

For them only a little more needed to be learned, and then all physical knowledge could be neatly sorted, packaged and put in the inventory to be drawn on for the solution of any human problem.

For one thing, there wasn't going to be any ceremony at all this year.

For that reason any democratic reform and effort to bring genuine representative government to the Dominican Republic will need the greatest sympathy and help.

For those communities which have financial difficulties in effecting adjustments, there are a number of alternatives any one of which alone, or in combination with others, would minimize if not even eliminate the problem.

For United States expenditures under subsections ( A ), ( B ), ( D ), ( E ), ( F ), ( H ) through ( R ) of Section 104 of the Act or under any of such subsections, the rupee equivalent of $200 million.

For the making of selections on the basis of excellence requires that any foundation making the selections shall have available the judgments of a corps of advisors whose judgments are known to be good: such judgments can be known to be good only by the records of those selected, by records made subsequent to their selection over considerable periods of time.

For the near term, however, it must be realized that the industrial and commercial market is somewhat more sensitive to general business conditions than is the military market, and for this reason I would expect that any gain in 1961 may be somewhat smaller than those of recent years ; ;

For any house.

For proper accreditation of schools, teachers in any course must have a degree at least one level above that for which the student is a candidate.

For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square.

For the lines of any plane, **yp, meeting Q in a conic C, are transformed into the congruence of secants of the curve C' into which C is transformed in the point involution on Q.

For any pencil in a plane containing a Af-fold secant of **zg has an image regulus which meets the plane of the pencil in Af lines, namely the images of the lines of the pencil which pass through the intersection of **zg and the multiple secant, plus an additional component to account for the intersections of the images of the general lines of the pencil.

For any choice of admissible policy Af in the first stage, the state of the stream leaving this stage is given by Af.

For in the modern world neither `` spirit '' nor `` matter '' refer to any generally agreed-upon elements of experience.

For in Christ Jesus neither circumcision nor uncircumcision but a new creation is of any account.

For example for any ( even infinite ) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks ( assumed to have no distinguishing features ), such a selection can be obtained only by invoking the axiom of choice.

: For any set X of nonempty sets, there exists a choice function f defined on X.

: For any set A, the power set of A ( with the empty set removed ) has a choice function.

: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.