The subsequence of the lower limit is monotonically decreasing because each term in the subsequence is the minimum value of the remaining elements in the original sequence. As we remove elements from the original sequence, the minimum value of the remaining elements can only stay the same or decrease. Therefore, the subsequence of the lower limit will continue to decrease or remain the same as we move through the original sequence. **

The subsequence of the infimum limit is monotonically decreasing because it is defined as the greatest lower bound of the sequence. As the subsequence progresses, it continues to find the greatest lower bound of the remaining elements in the sequence, which means it can only decrease or stay the same. Therefore, the subsequence of the infimum limit will always be monotonically decreasing as it finds the greatest lower bound of the sequence. **

When determining the convergence radius of a power series, we focus on subsequences because the convergence radius is related to the behavior of the terms in the series as n approaches infinity. By looking at subsequences, we can analyze the convergence properties of the series without being restricted by the behavior of the entire sequence. This allows us to identify patterns and make conclusions about the convergence of the series based on the behavior of its subsequences. **

No, the entire sequence is not necessarily a null sequence just because a subsequence is not a null sequence. A null sequence is a sequence that converges to zero, so if there is at least one subsequence that does not converge to zero, it does not necessarily mean that the entire sequence does not converge to zero. It is possible for the original sequence to still converge to zero even if there are non-null subsequences. **

No, finding a subsequence that is not a null sequence does not necessarily mean that the entire sequence is not a null sequence. A sequence can still be a null sequence even if it contains subsequences that are not null sequences. A null sequence is defined by the behavior of its terms as n approaches infinity, so it is possible for a sequence to have non-null subsequences but still overall converge to zero. **

In Islam, the concept of wasting money or being frivolous with wealth is discouraged. While physical coins have a tangible value and can be used for charitable purposes, virtual currency does not have the same physical presence. However, the principle of not being wasteful with wealth still applies. Therefore, throwing virtual currency, especially in a manner that is disrespectful or wasteful, would likely be considered haram. It is important to be mindful of how we use and treat our wealth, whether it is physical or virtual. **

In most countries, it is not forbidden to hoard currency in the form of coins. However, hoarding large amounts of coins may raise suspicion and could potentially be seen as an attempt to avoid taxes or engage in illegal activities. Additionally, some countries have laws that limit the amount of coins that can be used in a single transaction, so hoarding large amounts of coins may not be practical for everyday use. It's always best to check the specific laws and regulations in your country regarding currency hoarding. **

In numismatics, the inscription on a coin refers to the text or words that are typically found on the coin. These inscriptions can include the name of the ruler or issuing authority, the date of minting, and sometimes religious or symbolic phrases. Inscriptions on coins are important for identifying the coin's origin, historical context, and sometimes even its value. Studying these inscriptions is a key aspect of numismatics as it helps researchers and collectors understand the significance and history of the coin. **