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$\begingroup$ I was aware of Pascal using :=, but not the others.I think it is possible that the language designers of that time where influenced by maths (as has happened a number of times), but := is so far the only easily typable symbol mentioned here, so it is perfectly reasonable to assume it stems from programming languages in the first place.
In a (math) PhD personal statement/statement of purpose, should I use mathematical notation, or english, if math is likely clearer? How many years can a Boeing 747 be in service? What mnemonic helps differentiate 瑞士 (Switzerland) from 瑞典 (Sweden)?
When a sequence converges, that means that as you get further and further along the sequence, the terms get closer and closer to a specific limit (usually a real number).
However, a quick read through will provide some nice clarification. To that end, I suggest reading this excellent blog post on math.blogoverflow.com: More than Infinitesimal: What is “$\mathrm{d}x$”? that provides an excellent explanation of differentials. It does so by introducing differential forms.
All functions are well-defined; but when we define a function, we don't always know (without doing some work) that our definition really does give us a function. We say the function (or, more precisely, the specification of the function) is 'well-defined' if it does.
Here's a not too well-known instance of the use of $\varepsilon$ in mathematics: One somewhat well-known transformation for accelerating the convergence of a sequence is the Shanks transformation (after Daniel Shanks, who is probably more well-known for his number-theoretic contributions).
$$:=$$ is the commonest symbol to denote "is equal by definition." Note that $$\equiv$$ is used to denote an algebraic identity: this means that the equation is true for all permitted values of its variables. Rarely, however, it may denote a definition, so it's best to use this symbol only for congruences or identities.
A prerequisite to be able to answer the question "What does it mean to solve an equation?" is to have a definition of the word "equation". A definition is: an equation is an equality containing one or more unknown(s). Do you agree with this definition ? If yes, solving the equation consists either :
Sometimes "strictly between" is used to mean the version with strict inequalities. The word "between" should be avoided when making precise statements unless you explicitly clarify the meaning (or unless only one meaning could possibly make sense in context).
$\begingroup$ @Brian M. Scott What does it mean when each subscript is non-numerical, and is exactly the same? I have an equation that has a value T (temperature in Kelvin), subscripted gamma. It's used several times each the same way, no variation, and all other equations in the same family simply use T, unsubscripted. What might that mean ...